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The couple spoke to reporters, with Ardern holding her daughter, who was wrapped in a delicate white blanket and wearing a green woollen hat.
Ardern said she was doing well but was "sleep-deprived". She thanked hospital staff for her care.
The name Neve meant "bright" or "radiant", Ardern explained.
"We chose Neve because we just liked it, and when we met her we thought she looked like she suited the name.
"It means bright and radiant, and snow, which seem like a good combination for Matariki 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.
Ardern joked they'd "gone for simplicity" with the name "because Clarke with an 'e' has caused all sorts of problems, and Jacinda with a 'd'."
Gayford said the birth was "all a bit of a blur ... for both of us".
"Actually I won't forget the look on Jacinda's face when she finally held the baby. It was all a bit of a blur for the both of us, but she has obviously been through so much...she looked absolutely just stunned and very, very happy."
Ardern thanked members of the public who had sent in gifts or notes of congratulations.
"We'll see you all soon, and thank you again to the staff."
She also said it had been special to receive official messages.
"It's certainly mindblowing receiving messages from people like the Duke and Duchess of Sussex and the Queen, that's not something you'd expect. But it's equally special to us ... those people who took time to send a little note, or a blanket, or a set of booties."
A spokesman said the message Ardern referred to from Prince Harry and Meghan was a "lovely hand-written note" that looked as if it was written by Meghan.
"They congratulated the PM and Clarke and said they were looking forward to seeing them in the fall," the spokesperson said.
For the next six weeks while she was off work Ardern said she and Gayford would do what every parent did and figure out things as they went.
The first family spent three nights at Auckland City Hospital adjusting to parenthood since the baby was born at 4.45pm on Thursday. They will head home later today.
The couple announced the birth to the public on Instagram at 6.15pm that night.
No doubt Ardern will appreciate returning to her own home, after hospital meals including Marmite on toast, porridge - and her midwife's macaroni cheese.
Usually, new mums leave hospital soon after delivering and go to Birthcare in Parnell to recover.
But because of the security arrangements required for guarding the Prime Minister Ardern has remained at the hospital.
Ardern was driven to hospital by Gayford before 6am on Thursday. Almost 11 hours later, the little girl was born, weighing 3.31kg (7.3lb).
She has been described as having fine features, dark eyes, lots of dark hair and looking more like her TV fishing show host dad than her politician mum.
"She doesn't look like Jacinda as a baby because she's more tiny, more petite," Ardern's mother Laurell Ardern said.
Ardern has had little sleep over the past few days, with the baby also described as being "very alert and one hungry baby".
The baby is the first child for Ardern (37) and Gayford (40).
Gayford is a spear fisher and diver with his own television show. The couple have been together for about four years.
Since the birth the new family have since been inundated with good wishes, gestures and gifts, ranging from a private message from the Queen and congratulatory tweets from world leaders, such as Canada's Justin Trudeau and United Kingdom Prime Minister Theresa May, and a baby blanket from Ngāi Tahu.
The Embassy of the Kingdom of Saudi Arabia also sent a bouquet so large it could not fit in the Prime Minister's room in the maternity ward of Auckland City Hospital.
After the birth Ardern thanked everyone for their best wishes and kindness and said they were all doing really well thanks to the wonderful team at Auckland City Hospital.