The confirmation comes just a day after the transtasman bubble with Australia opened, allowing quarantine-less travel between the two countries.
Ardern told reporters in Auckland that today's case is unlikely to affect that bubble.
"These are the kinds of scenarios where we would anticipate movement continuing."
The Ministry of Health this afternoon revealed the border worker – who worked at Auckland Airport – had tested positive for the virus.
Ardern said the person cleaned planes that transported people with Covid-19 – "so a very clear link at this stage to cases that are high risk".
It is unclear at this stage which planes the worker was cleaning – Ardern would only say it was an aircraft that had arrived from a "high-risk" destination.
She said she has not spoken to Australian Prime Minister Scott Morrison about this case, but Health Minister Andrew Little has spoken to his Australian counterpart.
"When we opened, on both sides, we knew that we would continue to have cases connected to our border," she said.
"We accept that's going to be part of our journey together – Australia accepts that."
Aus Health Minister - 'full confidence'
Australian Health Minister Greg Hunt said this afternoon Australia has "full confidence" that New Zealand will be able to contain the new case of Covid-19.
Hunt said he has been briefed on the situation by his New Zealand counterpart.
"They're onto this. We have full confidence in New Zealand's system. We've seen them deal with the inevitable outbreaks, and there will be other days when there are cases in Australia," he said.
"We saw the outbreaks recently in Queensland and NSW – two states. And yet, we were able to deal with it.
"So we know how to deal with this. New Zealand knows how to deal with this."
The infected worker was fully vaccinated "quite early on in the campaign", Ardern confirmed.
The person was tested on April 12, as part of routine testing, and produced a negative result.
But, a week later on the 19, they had tested positive.
This shows the testing protocols were working as expected, she said.
"We still entirely expect that people who are vaccinated will still get Covid-19 – it just means they won't get [as] sick and they won't die".
Officials are now contact tracing the individual and will soon reveal any locations of interest.
Earlier today the Ministry of Health revealed that a new border-related Covid case has been detected on the second day of the transtasman travel bubble.
The ministry revealed Covid-19 had been detected in a worker at Auckland Airport.
It said the usual protocol of isolating the case, interviewing them, and tracing their contacts and movements was now under way.
More information will be provided later today.
Food-sharing initiative
Ardern is at Papatūānuku Kōkiri Marae in Māngere, paying a special visit to a community food-sharing initiative Kai Ika Project, that focuses on reducing waste.
The initiative aims to change the way people perceive "waste" through recovering unwanted fish parts and sharing them with people who want them.
A filleting service processes catch and ensures anything that isn't wanted is fully used.
In a Facebook post the project posted a picture of fish being smoked yesterday, saying it was preparing for a special visitor with a smoke-up of some fresh unwanted snapper and kingfish parts recovered from its filleting service.
"The biggest challenge with smoking fish is hiding it from the crew once it's done," it teased.
Asked about Ihumātao by the media, Ardern said it was not a botch-up by the Government.
She said the land will be utilised for housing but there is a "large process to go through" before the housing on the land is set up.
She said the funding was allocated as land for housing, but there is still a lot of work until the housing is built.
