This makes her the second elected leader in history to give birth while in office.
Ms Ardern was admitted to a hospital in Auckland early on Thursday morning, four days after her due date.
The 37-year-old has now passed on her duties to the Deputy Prime Minister Winston Peters.
She has said she will take six weeks of maternity leave.
Ms Ardern, who was elected in October, announced in January that she and her partner Clarke Gayford were expecting a baby.
"I am not the first woman to multitask. I am not the first woman to work and have a baby; there are many women who have done this before," she had said earlier this year in an interview with Radio New Zealand.
She is the youngest prime minister the country has had since 1856.
In 1990, Benazir Bhutto gave birth to a daughter while serving as Pakistan's prime minister, a first for an elected world leader.
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