More than 20 million people have now watched Video Games on YouTube; more than two million people worldwide own a copy of her album.
And there is much to like about the dozen torch songs and her smoky, not-quite-jazzy voice on Born To Die - especially, it appears, for the many, many women in the capacity audience of 1800 at the Palace.
If Del Rey's lyrics tend toward long-after-the-fact explorations of a brittle, lust-lorn adolescence or warmed-over teen angst, this doesn't seem to bother the mid-20s women standing next to me in Melbourne. They know all the words, gently singing along with the swoony, slightly loony enthusiasm of girls half their age.
They sway but they don't dance; what beats there are on her record are mostly absent for this performance. Del Rey has chosen to take her songs on the road with a string quartet, a bass player and pianist seated at a grand. The band, along with the large rubber plants around the side and back of the stage, the video loops featuring doomed, romantic figures like Elvis and the Kennedys and the melancholy stage lighting, do for mood while Del Rey - in bouffant hair, Alice headband, pleated, silver brocade dress and a necklace that catches the lighting - channels Jackie Kennedy, though I'm sure Mrs K would not have shown quite so much leg.
Del Rey's voice is strong, but her stagecraft is limited. Her image projects confidence, yet conversely her body language on stage seems hesitant, almost shy. More curious is that her image - and the images looping on the screen - seem out-of-sync with her music. Lyrically she's being bad - getting high with boyfriends or "stealin' police cars with the senior guys" - visually, this night, she looks like a goody-goody 60s prom queen with a thing for the Kennedys and Camelot.
Almost as I register my confusion over what this might mean, she's gone. Just 45 minutes after she wanders onstage behind her band, she slips off before it. There's no encore. Instead, her first night in Melbourne ends with the house lights going up to reveal faces that read: is that all there is?
This perhaps, explains some of the hate that came well-nigh immediately after the first flush of love. Almost as soon as Del Rey was discovered by the world wide web last year and electronically anointed as the next hot, homemade, indie music It-girl, it was also discovered that she was no overnight success at all. Instead, the internet's trolls tumbled that she'd been performing in small clubs and open-mic nights in New York for years and that her previous crack at the music business had been little more than a year before Video Games was uploaded to YouTube; in 2010 she released an album that sank with all hands aboard and a few years before that had recorded an album that was not even released.
And, horrors, her name wasn't even Lana Del Rey. It was Elizabeth "Lizzy" Grant, and she was, despite her hot stuff video clips, no LA bad girl but is instead a middle-class, boarding school and college-educated daughter of an investor born in 1986 in Lake Placid in upstate New York.
From here, the rumours became more rancorous and speculation more paranoid: she has collagen lips; she was a rich girl on an ego trip funded by her daddy; her image and possibly her songs were created for her by an army of evil record company advisers; someone, possibly the evil record company advisers, had attempted to hide or delete all references to Lizzy Grant on the web; she is the latest "Illuminati puppet"; that - in the wake of a semi-disastrous appearance on American sketch show Saturday Night Live in January - she can't even sing ...
Then there was the sheer nastiness, such the blogger who raged "before she was alternative, she was a failed mainstream artist without fake lips ... She had blond hair, didn't look very 'alt sexy'. Sorta like a girl from my high school who was a part-time hostess at [US restaurant chain] Chili's." Ouch.
Inevitably, when Del Rey has been interviewed by the mainstream media, it has been the so-called controversy over her authenticity rather than her music that's dominated the questions. It has to be said that she hasn't exactly front-footed the issue.
So I am unsurprised that, when I finally broached the subject, her answer was opaque; throughout my 20-minute audience her answers to questions were by degrees evasive, sometimes inscrutable and, at least now-and-then, sounded a little bit mad.
Lying back on a couch in her Melbourne hotel room, dressed in tight blue jeans and a lemon cable-knit jumper, with her auburn tresses tied up in a rough knot atop her head, her feet bare and her face not made up, she looked more Lizzy Grant than Lana Del Rey. Only a diamond (or diamante) necklace hinted at her public image.
She was utterly polite in that way Americans are, but with little warmth. She was at her brightest after the dictaphone was switched off and we exchanged brief thank yous and farewells. She was at her coolest when I asked how she's coped with the internet-led opprobrium. Was she hurt by the backlash?
"I mean, well, I don't know why it would be interesting to know whether I was hurt or not, so I don't know ..."
Well it seemed rather intense and uncalled for, I suggested.
"I think the only thing that I would say about that is that I agree with you. And yeah, sure, it is uncalled for. But then again hatred always is uncalled for because it is just not right to live that way. Huh ..." She said "huh" like the thought had just occurred to her and it pleased her.
If the backlash has been gratuitous and, at times gratuitously offensive, this seems more about the puritanism of internet music bores who thought they discovered for themselves some rare and exotic creature only to realise, horrors, that she's just another in a long line of pop singers - David Bowie, Elton John, Gaga to name a few - who has found success after a name and image change.
The question of course is whether Lana Del Rey is a stage name or more than a stage name.
"It's kind of less than a stage name."
Ah, okay. So what is it?
"It's just a different name. But you know I got to know myself so well so many years ago when I got sober and I did kind of start learning about the right way to live and the things that made me happy - which was to create and other things. And so I've always been the same person that I am now and never did things that regularly. I always knew I'd have my own name, I knew I'd have my own world and ... but, you know, if I was in the same room as you and the people I've only worked with for the last 12 months and I was also in the room with my younger brother or my teacher from second grade no one would ever notice the split in personality. It's all very much consistent, there would be no surprises."
Would it matter if Lana Del Rey was an invention, a music biz contrivance? Not at all. Or at least it shouldn't. The songs are the songs, her voice is her voice, the pimped image is the pimped image. Take it or leave it.
However, it's not quite that simple. Though she doesn't quite say it, it she does seem to be playing the part of singer to some degree. While she feels at home in the recording studio she is, she admits, shy when in front of audience.
"I think, you know, it is a short life. It is important to show yourself in the light that you'd like to be shown and the light I'd like to be shown in is not necessarily in a spotlight in front of everyone else. I'd love to introduce myself to people through my words and the way that I think because I like the way that I think. The way that I look on stage in front of thousands of people is not really my thing but I do my best."
And she's having to do her best in front of audiences that have grown in size from dozens to hundreds to thousands in the last year. Her manager, Ed Millett, who oddly enough is English not American, says part of the reason Del Rey has, between endless promotional duties, been performing with strings, piano and bass rather than a full band is that it would have been too costly for her level of popularity at the beginning of year, but also it meant no sound check was required before a show.
"She hasn't done festivals before, it's her first year and there's a lot of attention on her. So going onto a stage with no sound check in front of 10,000 people with a big drumkit, smacking beats when you're not used to it, is like a whole new thing."
Millett says a world tour with full band is on the cards for next year - after a special edition of Born To Die has been released.
"She hasn't really been on the road [this year] but it is a combination of promotion and touring. She's been doing promo since January. She's done little spurts of one-off shows in the gaps between. Usually you do your home market and you get that up and running and then you go out. But because there was so much heat on the record, it was a worldwide release, she's basically not had [a break], she's had to kind of go around the world 50 times ... so it's been quite epic"
Del Rey says she's tired all the time. However, the special edition of the album requires more songs, but that's okay because she's always loved to write, apparently.
"I was writer for the last 20 years," says the 26-year-old. Indeed it is writing, rather than music or performance, she loves the best, something she likes to do alone in a room "with the windows open, that's my process."
What she tends to write about, if the lyrics on her album are a measure, is herself, but herself as a teenager. She seems to have been - though she won't discuss this with the media - something of a bad girl in her teens, so bad that she had to "get sober" a decade ago.
How bad did it get? "Just to the point where I appreciated the feelings that it [alcohol] gave me more than the feelings that I got from being with anyone else I knew personally. That's a problem, that's a problem."
Her sobriety seems to be front of mind always, it certainly keeps coming up in her conversation, and her thoughts around it sound like they've been formed in therapy. Not drinking is about "living your life without lies", she says, and she now has "nothing to own up to because I don't have anything I do wrong."
However, 10 years on from sobering up she still has to make sure she's not around bad things or people. "It's scary. You can feel it happening. You tell yourself that it is just the way life is, but it's not."
If her wild years have given her lyrical inspiration, I wonder out loud whether the bubble of touring, promotion, hotels and performance that she is now in might mean she finds little new to write about in the future. Apparently not.
"My mind was always a really interesting neighbourhood to live in. [It is] the way I was wired when I was born. [Crime writer] James Ellroy used to say like he, you know, had enough crazy thoughts in his head to sustain him for the next 40 years, all he needed was quiet. That's all I need too, right now. I've done a lot and I don't need to do anything else in terms of life experience."
Except perhaps learn to cope with her growing celebrity. Del Rey seems to be as mystified by her rise from obscurity as her detractors.
She had been making music for years, and made and uploaded videos for every song on her second, failed album, with no success. Then she made Video Games. What changed? Well, she has no idea.
"Well, I don't ... ummm ... dwell on it because nothing ... I have no idea what is going on. Like my situation is unpredictable. The only thing consistent in my life is myself and the way that I think. I live in a world where I am the only one who is doing ... you know ... who is consistent."
When I ask what kind of temperament she has, she replies - instantly - "peaceful". But she sounds a long way from peaceful when I ask what it is she wants from music. She sounds a little confused, certainly uncertain.
"I don't think it [my success] is a very sure thing at all ... But I don't want anything from anything. I got what I wanted a long time ago, which was peace, when I decided that I would write just to write, because it was my one love. And to be a good person [she says this in that high, emotional tone you hear in teenagers] and to be with my family and friends ... it's ... it's ... I haven't wanted for anything in many, many, many years, not in any kind of way.
"I don't really feel that deep desire for things to expand or for things to change. I got what I wanted when I found myself through writing and I've been happy ever since."
