The article, by author Anna Pasternak, refers to the eating disorders suffered by William's mother Princess Diana, saying: "Kate has become perilously thin, just like – some point out – Princess Diana."
A royal source told the Mail on Sunday: "That is such an extremely cruel and wounding barb. It's disgusting. It's sexist and woman-shaming at its very worst.
"The piece is full of lies. There is no truth to their claim that the Duchess feels overwhelmed with work, nor that the Duke is obsessed with Carole Middleton. It's preposterous and downright wrong.
"The whole thing is class snobbery at its very worst. The stuff about [Kate's sister] Pippa is horrible. Tatler may think it's immune from action as it's read by the royals and on every coffee table in every smart home, but it makes no difference.
"It's ironic that the royals' favourite magazine is being trashed by them. The Duchess is a naturally shy woman who is doing her best."
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Tatler's article also claims that Kate, 38, had a spat with Meghan Markle before her wedding to Prince Harry in May 2018.
It says the clash took place at a rehearsal two days before the big day and centred on whether Princess Charlotte and the other young bridesmaids should wear tights. Quoting an unnamed friend, the article says: "There was an incident at the wedding rehearsal. It was a hot day and apparently there was a row over whether the bridesmaids should wear tights or not.
"Kate, following protocol, felt that they should. Meghan didn't want them to. The photographs suggest that Meghan won."
But Kensington Palace insists the story is wrong.
Instead, the Mail on Sunday understands the fallout was over Meghan's reaction to Kate's request that the hem of Charlotte's dress be lengthened. The resulting row left Kate in tears.
The profile takes aim at Kate's taste and her family, sneeringly describing the Cambridges' Anmer Hall home in Norfolk as "very Buckinghamshire".
"Carole has put her stamp on Anmer decor-wise," it claims. "Far from being a typical aristo abode, with threadbare rugs and dog hair everywhere, like, say, Windsor and Balmoral, it is, according to a visitor, 'like a gleaming five-star hotel, with cushions plumped and candles lit'."
It also labels Mrs Middleton "a terrible snob" and snipes at Kate's younger sister Pippa as "too regal and try-hard ... A bit lost now and is struggling to find her place".
To the anger of Kensington Palace, Pasternak asserts that the Cambridges feel very tired because they have been forced to take on more royal duties after Megxit.
The article cites a source as saying: "Kate is furious about the larger workload. She feels exhausted and trapped. She's working as hard as a top CEO, who has to be wheeled out all the time, without the benefits of boundaries and plenty of holidays."
But figures in the Court Circular, the official record of royal engagements, suggest otherwise. The couple have done a similar number of jobs from January to March as they did over the same period last year. Kate has actually done fewer – 29 this year against 35.
After the article was published, Kensington Palace said: "This story contains a swathe of inaccuracies and false misrepresentations which were not put to Kensington Palace prior to publication."
Hitting back, Tatler said: "Tatler's editor-in-chief Richard Dennen stands behind the reporting of Anna Pasternak and her sources.
"Kensington Palace knew we were running the 'Catherine the Great' cover months ago and we asked them to work together on it. The fact they are denying they ever knew is categorically false."
The Mail on Sunday understands Tatler approached Kensington Palace in February to ask if Kate would pose for their cover or provide an exclusive cover photo.
That request was politely declined and it is understood Kensington Palace was not offered the chance to comment on the specific content of Pasternak's article.
Tatler did not respond to a Mail on Sunday request for comment and Kensington Palace declined to comment.