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Neve Te Aroha Ardern Gayford made her way down the Auckland Hospital corridor in her mother Prime Minister Jacinda Ardern's arms, flanked by her exhausted but proud-looking father, Clarke Gayford.

Ms Ardern left Auckland City Hospital this morning with partner Clarke Gayford by her side.

The island nation, with a population of 4.5 million, was in the grips of a baby-induced glow of optimism in recent days with many social media users dubbing the baby the "Prime Miniature" while they waited for her name to be announced.

"Also it means, in various forms, bright and radiant and snow, which seemed like a good combination for Matariki (Maori new year) and for solstice".

"Te Aroha is our way of reflecting the amount of love this baby has been shown even before she arrived.it's also the place where all my family are from and I grew up under that mountain", she said of the Waikato landmark.

And baby makes three!

Ardern's partner Clarke Gayford spent Thursday night with Ardern at the hospital.

Ms Ardern said she and her partner had opted for the alternative spelling Neve rather than the traditional Niamh, to avoid confusion.

The couple on Sunday described to reporters the first moment they met Neve.

"The baby was looking up at Jacinda and it looked like she was in awe of her and I couldn't get over how alert it was just after being born", she said.

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Medical providers who have visited the facilities say they are safe and clean, but the children are often crying and acting out. Blame is also placed on the parents for putting their children in this position by illegally crossing borders with them.

"But then I was a bit delayed obviously and it just so happened that Jacinda was a bit delayed as well, meaning we had our babies on the same day after all".

A heavily pregnant Jacinda Ardern handles the cut and thrust of parliamentary debate.

The couple announced the birth of their baby girl Thursday on Instagram, saying they feel "very lucky".

"We chose Neve because we liked it and when she arrived she looked like a Neve", says Ardern.

The family will now leave the hospital to spend a stint at home and away from the spotlight.

Many in New Zealand and around the world had held up Ardern's pregnancy - just the second time an elected leader had given birth in office since Pakistan's Benazir Bhutto in 1990 - as a symbol of progress for women.

Ardern previously said she plans to return to work at the beginning of August.

The pregnancy news late a year ago came as a surprise to Ms Ardern, who then had to simultaneously deal with both intense negotiations to form a government and morning sickness.

"I hope for little girls and boys that actually there's a future where they can make choices about how they raise their family and what kind of career they have that are based on what they want and what makes them happy".