The grandmother of the First Baby has described holding the newborn for the first time.

"It's unbelievable, just lovely," Laurell Ardern said of her daughter, Prime Minister Jacinda Ardern's, two-day-old baby.

"She's such a little cutie and with such fine features."

The baby has lots of dark hair and has dark eyes but doesn't look like her mum, Laurell Ardern said.

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The little girl looked more like her father, Clarke Gayford.

She hadn't given her daughter advice, Laurell Ardern said.

"She's the Prime Minister. I don't need to give her any advice. Jacinda works things out for herself."

She didn't know who had changed the first nappy or given the first bath. She had planned to ask Gayford if he'd changed the baby's nappy but had forgotten.

"I'm sure he has."

The pair would be great parents, the proud grandmother said.

"Clarke is so easygoing and Jacinda is very caring and loving."

Soon after the baby's birth, Laurell Ardern asked her daughter about the moment the baby was born.

"She said it was just lovely, and I could hear in her voice [what it meant to her]."

She had not asked Gayford the same question, but she did not need to.

"I can just see it in his face. He's very proud."

She "didn't know anything" about the baby's name, Laurell Ardern said.

But she reported again what others had said - that the baby was sleeping well and was very alert.

"She's looking around."

The Prime Minister was doing well too, her mother said.

"She looks great."

It was announced this morning that Ardern will leave hospital with her newborn daughter at 11am tomorrow.

The name of the baby, born on Thursday, is yet to be released.

The baby slept well last night but Ardern did not fare so well, media were told.

An 'alert but very hungry baby' has meant Prime Minister Jacinda Ardern won't be making any public appearances today but that hasn't stopped well-wishers, including the Queen, from sending thousands of congratulatory messages.

The baby will be given her first bath this afternoon and a new photo of her was expected to be released soon.

The PM had porridge for breakfast and had opted to stay in hospital for a third night.

Usually, new mums leave hospital soon after delivering and go to Birthcare in Parnell to recover.

Because of the security arrangements needed by the Prime Minister and Gayford, she has had to stay at the hospital.

Media have been camped out at a room at Auckland Hospital since the PM arrived in labour early on Thursday morning.