The baby girl is the first child for the 37-year-old and her 40-year-old partner Clarke Gayford, with the youngster weighing in at a healthy 7.3 pounds (3.3kg). The prime minister's office brought food for reporters waiting at the hospital.
There was no mention of a name, although Ms Ardern earlier warned the couple were struggling to pick one.
This makes her only the second elected leader in modern history to give birth while in office.
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Ms Ardern arrived at the country's largest public hospital at 5:50am (local time) with Gayford, her spokesman said.
Church said: "She's going to be very busy for those six weeks with the baby and I suspect she's going to be doing a lot of hard work at home as well".
However, the Prime Minister will still be consulted on all major decisions.
Ms Ardern had played down the chances of any disruption arising from her absence, saying she and Mr Peters would be in regular contact.
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"Like the rest of the country we welcome the news of the birth of a healthy baby and are naturally delighted for the new parents".
New Zealand's parliament has baby-friendly policies with the Speaker of the House Trevor Mallard once looking after the child of a Labour MP while the country's parental leave bill was being debated, Stuff.co.nz reported.
The first female prime minister was Sirima Bandaranaike, who was elected to lead Sri Lanka (then Ceylon) in 1960. One group started a campaign, #KnitForJacinda, to knit baby clothes for hospitals and women's shelters in celebration of Ardern's news.
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