"Thank you", Ms Ardern said in a statement.
"Welcome to our village wee one."
Former New Zealand prime minister Helen Clark sent her well wishes via Twitter, saying it was a proud day for her country and emblematic of a "21st New Zealand".
Ms Ardern posted a photo on Instagram of her holding the new baby with partner Clarke Gayford.
Jacinda Ardern and Clarke Gayford arrived at Auckland Hospital this morning to have their baby.
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Anyone will be able to launch an IGTV channel, leaving the door open for a community of homegrown creators to arise on the app. It has been largely insulated from controversies, such as Facebook's data privacy woes or YouTube's problems with curation.
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Later, he declared during remarks at a hotel that he had no choice but to remove children from their parents at the border. In the seven-minute tape, children who were reportedly aged four to 10, could be heard crying out for the parents.
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He'll also be seen in the Darwin Strike League in July where he will play for Randwick Petersham, his Sydney-based boyhood club. Bancroft is still contracted to Western Australia and Perth Scorchers and is set to join them for pre-season training in June.
Mr Gayford will become a full-time dad, while Ardern continues to lead the country after her six weeks scheduled leave. "Her and Clarke have my best wishes for this special experience".
"The birth has been highly anticipated in the South Pacific nation of almost 5 million people, and Ardern has continued to work right up until the birth, with her partner posting on social media on Wednesday afternoon a photo of her poring over official documents from home with the hashtag "#stillwaiting". On Thursday, many major media outlets set up livefeeds on their websites with rolling coverage and local journalists camped at the hospital.
In 1990, Benazir Bhutto gave birth to a daughter while serving as Pakistan's prime minister, a first for an elected world leader.
Ms Ardern addresses parliament in Wellington.
She said she hoped to be "sharing the good news" in an announcement but also to have some quiet time to enjoy as a family. Gayford, a television presenter, will become "first bloke" and a stay-at-home father to the child.
Peters, head of the populist New Zealand First party, had already taken over some of Ardern's duties, including running cabinet meetings in the capital Wellington while she stayed close to home in Auckland.