Ardern, 37, and her partner Clarke Gayford announced their first child's name as they left Auckland Hospital on Sunday.

Ardern said the new family was "sleep-deprived but super well" as she thanked hospital staff and members of the public for their support.

"We chose Neve because we just liked it and when we met her we thought she looked like she suited the name.

"Also it means in various forms 'bright' and 'radiant' and 'snow'."

Te Aroha - the Maori-named rural town in the Waikato region of New Zealand that is often referred to as the place of love - was where Ardern's family was from, she explained.

"Te Aroha ... was our way of reflecting the amount of love this baby's been shown before she even arrived."

Neve Gayford, as her mother said she will be known, wore a green knitted hat and slept through her first media appearance.

"We're not placing any great expectations on this little baby except happiness and love," she said.

Gayford said the birth was "a bit of a blur for both of us" but he would not forget the look on Ardern's face when she first held her daughter.

"She had obviously been through so much and the moment had arrived and she looked absolutely just stunned and very, very happy."

In a video posted to her Facebook page, Ardern poked fun at her partner's choice of a grey knitted cardigan as the family prepared to leave their hospital room.

"Clarke's got his dad-cardi on, so that means he's ready to go," she said, to which Gable replied the op-shop find had finally "come of age".

She wrote the pair "just wanted to say thanks, from the bottom of our hearts" as they headed home.

Ardern shared news of the baby's Thursday afternoon arrival on her Instagram account last week, writing "welcome to our village wee one".

"Feeling very lucky to have a healthy baby girl that arrived at 4.45pm weighing 3.31kg (7.3lb) Thank you so much for your best wishes and your kindness. We're all doing really well thanks to the wonderful team at Auckland City Hospital."

Ardern became New Zealand's youngest prime minister when she took office through a coalition deal last year after an inconclusive election.

She is the first woman in the country's history to give birth while in office.

Deputy Prime Minister Winston Peters has stepped in as acting PM and will run the country for the next six weeks while Ardern takes maternity leave, according to an agreement they published earlier.

Ardern is one of the few elected leaders to hold office while pregnant. Pakistan's Benazir Bhutto gave birth while she was prime minister in 1990.

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

Then, Gayford will take care of the baby and will travel with Ardern between their Auckland home and the capital, Wellington, as well as on international engagements.

Former New Zealand prime minister Helen Clarke congratulated the pair and said on social media their parenting arrangement was "gender equality in action. This is [21st century New Zealand]."

Ardern worked until late into her pregnancy, regularly encountering members of the public who touched her stomach and passed on gifts such as 'onesies' and miniature gumboots.

"It's been great," she told reporters at her last major public event before giving birth.

"New Zealanders are incredibly generous people and have been generous in their support of me regardless of the politics just as another human being going into a new stage of life."

with Reuters