Ms Ardern, 37, became New Zealand's youngest prime minister when she took office through a coalition deal previous year after an inconclusive election.
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. Ms Ardern later broke news of the birth on social media, saying she felt very lucky and thanking staff at the hospital.
He said Ardern should be allowed to acclimatise to motherhood and not be held to higher standards because she was prime minister.
The 37-year-old will take six weeks of maternity leave, and has now passed on her duties to Deputy Prime Minister Winston Peters.
This has all been a new challenge for her staff as they try to balance the inevitable domestic and global media demands, while trying to make the birth as "normal" as possible for Ms Ardern and partner Clarke Gayford.
Interestingly, Ms Ardern's baby shares the same June 21 birth date as the former Pakistani leader, who died in 2007.
The first woman elected to Britain's House of Commons was Constance Markievicz in 1918, although as a member of the Irish republican party Sinn Fein, she did not take her seat.
That may include a disciplinary matter - all Cabinet ministers serve at the pleasure of the Prime Minister regardless of the party they're with.
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She is the first world leader to give birth in almost three decades. "I'll be Prime Minister AND a mum, and Clarke will be "first man of fishing" and stay at home dad".
The Prime Minister will take six weeks off.
Under the arrangement, Ardern will still be consulted on major decisions, including issues of national security.
When announcing the pregnancy, Ardern said she and Gayford were surprised they were able to conceive naturally.
Ardern's partner, Clarke Gayford, will be a stay-at-home father.
Ardern is NZ Labour's youngest ever leader and the country's youngest PM since 1856.
"Like the rest of the country we welcome the news of the birth of a healthy baby and are naturally delighted for the new parents". Women can choose to combine family and career as Ardern has done.