Ms Ardern spoke briefly to reporters today before heading home, revealing her and her partner Clarke Gayford had named their daughter Neve Te Aroha Ardern Gayford.
Ardern is one of only a few female leaders to give birth while holding office - the most recent being former Pakistani PM Benazir Bhutto, who had her daughter in 1990.
"It was a real find, anyway", Ardern said.
"Maybe you've all had a calming effect as she's finally gone to sleep", says Ardern, as she fronted a media scrum as the anxious sound of camera shutters fluttered in the background.
Te Aroha - a rural town in the Waikato region of Zew Zealand - was where Ardern's family was from, she explained.
"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".
"And that it's generally accepted, not just that women can make choices, but actually that men can too", Ardern added, referring to Gayford, who was standing beside her.
"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". Te Aroha was something we settled on quite early.More news: Critics Rip NYT for Killing Audio Interview With Stephen Miller
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Deputy Prime Minister Winston Peters is filling in for Ardern while she is on parental leave.
Ardern delivered her 3.31 kg (7.3 lb) daughter in the country's largest public hospital on Thursday afternoon with her partner, television presenter Clarke Gayford, at her side.
The little baby girl, who is absolutely divine, as a very special and attractive name. "So, I'm dying to see it and hold her and just see what's she's like".
Ardern is admirable because she has reached the pinnacle of leadership, without sacrificing personal joys like marriage and parenthood.
"Not only is she incredible at what she does, this morning she made me macaroni and cheese because she heard me mention a wee craving yesterday", Ardern wrote.
It's the couple's first child together, and comes eight months after the 37 year old was sworn in as Prime Minister.
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.
"It's a real pleasure to introduce our little one to you all and to New Zealand", Ardern said on Sunday, thanking both hospital staff and members of the public for their "kindness, warmth and care" over the course of her pregnancy.