Gayford tweeted the same photo of the happy couple and their newborn baby on Twitter, saying: "So happy to announce our little girl has finally arrived!"
The birth capped an eventful year for Ardern, who became prime minister in October just three months after taking charge of the Labour Party as it languished in the polls.
The New Zealand leader said she was blown away by well-wishes locally and internationally, including from Britain's Queen Elizabeth II and the Duke and Duchess of Sussex, Prince Harry and his wife Meghan.
Ms Ardern will be the first world leader to take maternity leave.
"We're not placing any great expectations on this little baby except for happiness and love", she said.
"Te Aroha was our way of reflecting the amount of love this baby has been shown before she arrived and all of the names we were gifted along the way (by various iwi - or tribes)", Ms Ardern said. She previously said that she was able to do everything because she had "enormous support around me and it makes me quite privileged", refusing the impression that women had to be super human to do all.
Ardern answered questions with a broad smile while rocking Neve back and forth as she slept. It's such a rare event, in fact, that the last time anyone can remember it happening anywhere was in 1990, when Pakistan's then PM Benazir Bhutto popped out a baby. Deputy Prime Minister Winston Peters has taken the reins as acting prime minister.
"We chose Neve because we just liked it, and when we met her we thought she looked like she suited the name".
"So I hope for little girls and boys that there's a future where they can make choices about how they raise their family and what kind of career they have, that are just based on what they want and what makes them happy", she said.
The name, which has Irish origins, is traditionally spelt Niamh and variously means "bright", "radiant" and "snow", Ms Ardern said.
She will take six weeks of leave before returning to work.
"I'll be Prime Minister AND a mum, and Clarke will be "first man of fishing" and stay-at-home dad", the 37-year-old wrote.
Former New Zealand Prime Minister Helen Clark said the birth and the way the country had greeted it would be seen as inspirational by advocates for gender equality and women's empowerment.