Prime Minister Jacinda Ardern's political colleagues have offered the couple their congratulations after the birth of their daughter. Ardern has yet to announce the name of the child.
"At this meeting the Kaiwhakahaere, Lisa Tumahai, Deputy Kaiwhakahaere, Matapura Ellison and Chief Executive Officer, Arihia Bennett, gifted the Prime Minister elements of our Ngāi Tahu pēpi pack for her baby".
Ardern had little sleep last night, after celebrating her daughter's birth late yesterday afternoon with a dinner of Marmite toast and Milo last night.
Former New Zealand Prime Minister Helen Clark believed Ardern and Gayford had sent a significant message to the world, particularly with Gayford being a stay-at-home dad. Jacinda Ardern is the second female prime minister of the world who has been a mother while in the post of a Prime Minister.
In a recent interview, Ms Ardern said she was bracing for life in a media bubble for her family.
Asked about which surname would be given, Ms Ardern in January said she expected the baby to be a Gayford.
New Zealand must be so very, very proud of the wee one that's entered its village.
Tributes have come in from around the world, including from Britain's Queen Elizabeth II and from Google, which posted an image in the shape of a heart on its New Zealand homepage along with the message "Congratulations!"
Her plans for a family had sparked a sexism row during the election when a television host quizzed her on the issue, saying voters had a right to know before they cast their ballots. "Based on what I've heard so far she seems very alert - just like my daughter Isabella who is very excited to have a baby cousin", she said. "It's inspirational for younger women and men". The last head of government to give birth while in power was Benazir Bhutto, who had her second child in 1990, while prime minister of Pakistan.
"It is a woman's decision about when they choose to have children and it should not predetermine whether or not they are given a job or have job opportunities", she said then.
Ms Ardern plans to take six weeks leave from Parliament. She said that in other fields women have been combining motherhood and paid work for decades, but it has only recently become more manageable thanks to paid parental leave.
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