"I'm sure we're going through all of the emotions new parents go through, but at the same time feeling so grateful for all the kindness and best wishes from so many people".
Bakhtawar tweeted "congratulations" to Ardern after the announcement, and shared a link to a news story on how Bhutto showed it was possible to be a mother and a prime minister.
"Lots of people juggle a lot of things in their personal and private lives, and I'm not unusual in that". "We're all doing really well", she wrote.
"The baby was looking up at Jacinda and it looked like she was in awe of her and I couldn't get over how alert it was just after being born", she said.
Ardern and Gayford, who hosts a television fishing show, had known the sex of the baby for several months but made a decision to keep it secret from the public.
Some of the well wishes on Prime Minister Jacinda Ardern's Instagram, including messages from Helen Clark and pop singer Pink.
Before Ardern, former Pakistan Prime Minister Late Benazir Bhutto was the only woman to have undergone labor while in power.
She said the trick to making it work as a family unit was having a supportive partner, appreciating your partner and switching off from work when it was family time.More news: United Nations chief regrets as U.S. exits 'biased' Human Rights Council
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"The term Prime Miniature is just too flawless. please can we call her nothing else?" a person called Nic tweeted.
Meanwhile Gayford, in committing to be a stay at home father after Ardern finishes up her six weeks parental leave, is also shattering assumptions and offering a much-needed high-profile reminder that men can take on this role.
New Zealand resident Jenni Werth, who is pregnant, told The Guardian that she is "intrigued with how much of a difference [the pregnancy] might make in her understanding of what it's like to be a working mum - and she's obviously at one extreme of the spectrum".
As well as national and global joy, the news led to an awkward discussion for these parents.
Acting Prime Minister Winston Peters led the well wishes from the Government, saying it's a "happy day", along with members of the opposition and those within the Prime Minister's own party, including Finance Minister Grant Robertson.
"Earlier this morning my office was advised the Prime Minister was being admitted to hospital in preparation for the birth of her child".
In an interview with Radio New Zealand, she said, "I am not the first woman to multitask".