Finance Minister Grant Robertson says the Prime Minister had not yet named her baby.
Jacinda Ardern was admitted to the hospital at 5:50 am on Thursday morning and the baby was born at 4:45pm.
She confirmed the birth on Instagram, with a touching photo of her, partner Clarke Gayford and her baby girl.
Mr Robertson told the AM Show he has been in contact with Ms Ardern since the birth to congratulate her and the Prime Minister and her partner have been overwhelmed by the outpouring of support.
"I've exchanged messages with Jacinda and that's the main thing that she's said, she's just really grateful for the outpouring of love," he said.
"I feel like New Zealand's the happiest country in the world this morning, it's great to have that feeling where everyone gets together.
"There's 4.8 million aunties and uncles out there, I think Jacinda and Clarke are really feeling that love right now."
The couple still haven't chosen a name though and Mr Robertson says they're thinking about what will best suit the little girl.
"Obviously like most people they think about family names, names that are special to them, you know she's a beautiful little girl," he said.
"I'm sure they'll look hard at her and see what names seem appropriate but uh, no doubt we'll hear about that soon."
Later in the program MP Judith Collins and AM Show host Amanda Gillies suggested their own names for the baby.
Ms Collins said she thinks Amanda Anne would be a good name for the baby, while Gillies said the child could be named Helena.
Mr Robertson praised the couple for having their baby at a public hospital, saying it said great things about New Zealand.
"Isn't it great that in New Zealand the Prime Minister has her baby in a public hospital?" he said.
"That she's driven there by her partner in their car and along with what 20 odd other women who had babies at Auckland hospital yesterday.
"I think that says some really good things about our country."
Newshub.
