New Zealand Prime Minister Jacinda Ardern and her partner Clarke Gayford have introduced the public to her daughter and revealed the newborn's name.

The beaming pair greeted media on Sunday for the first time since arriving at Auckland Hospital on Thursday.

Ms Ardern began by announcing the baby would be called Neve Te Aroha Ardern Gayford — or Neve Gayford for short.

"When we met her we thought she looked like she suited the name," she said.

"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."

Aroha in Maori means love and Te Aroha is also the name of a mountain near where Ms Ardern's family come from.

"Te Aroha was our way of reflecting the amount of love this baby has been shown before she arrived and all of the names we were gifted along the way [by various iwi — or tribes]," Ms Ardern said.

Clarke Gayford will be Neve's primary carer once Ms Ardern returns to work in six weeks' time. ( AAP: David Rowland )

She and Mr Gayford had opted to spell the name Neve — as opposed to Niamh for example — for simplicity, with Ms Ardern joking about the unusual spellings of Clarke and Jacinda.

The couple also described the first moment they met Neve.

"I won't forget the look on Jacinda's face when she finally held the baby," Mr Gayford said.

"It was all a bit of a blur for the both of us … She [looked] absolutely just stunned and very, very happy."

Since the birth on Thursday, Ms Ardern has received messages from world leaders, including the Queen, as well as the Duke and Duchess of Sussex — who sent a handwritten note saying they looked forward to meeting later in the year.

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"As equally special to us were just those people who took time to send a little note, or a blanket or a set of booties," Ms Ardern said.

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

"Over the next six weeks we'll do what every other parent does and learn the little nuances … and just figure things out as we go."

Before leaving the hospital, Ms Ardern recorded a personal video, with a sleeping Neve in her arms that was posted on her Facebook page, to say "thanks, from the bottom of our hearts" for all the messages and comments they have received.

She also again praised the staff at the hospital for the care, advice and support she received during their hospital stay.

"We couldn't have had a better start," she said.

Ms Ardern has become the first elected world leader to take maternity leave, but for the next six weeks will still be on call for major government issues.

Mr Gayford will be the child's primary carer once the PM returns to parliament.

"Clarke's being as much of a role model here as I am," Ms Ardern said.

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

Jacinda Ardern continued to work right up until her baby's birth. ( AP: Nick Perry )

AAP