Gordon and Sarah Brown wed in August last year
The girl, who has not yet been named, was born in Scotland by Caesarean section on Friday, seven weeks early.
She weighed just 2lb 4oz (1kg) but is breathing independently, and mother and daughter are said to be "doing fine".
Prime Minister Tony Blair was among those to send congratulations to the couple, from Egypt where he is on holiday.
Dr Tahir Mahmood, the consultant obstetrician who delivered the baby, said Mr Brown cuddled his new daughter after she was born.
The doctor made the decision to operate following a routine scan on Thursday and consultation with colleagues and the parents.
Risk
He said the baby was revealed to be suffering from intra-uterine growth restriction, a condition which occurs in 10% of pregnancies and causes the child to have an irregular heart rate.
Mr Brown was at his wife's bedside for the birth at 1216GMT at the Forth Park Hospital in Kirkcaldy, Fife, near his Dunfermline East constituency home.
His spokesman said: "Both mother and baby are doing fine. Gordon and Sarah have not yet decided on a name.
"They are both delighted with their beautiful baby girl.
"Sarah is recovering well and will go home after the weekend, but, as normal, the baby will remain in hospital to make sure everything remains fine.
"They would like to thank the doctors and nurses and all the staff at the hospital for all their wonderful care."
Mrs Brown was brought into the theatre on Friday morning, when a team of two obstetricians, including Dr Mahmood, one paediatrician, one anaesthetist and three nurses delivered the baby in a Caesarean operation which took 30 minutes.
'Father's looks'
Dr Mahmood said that if the pregnancy had been allowed to run its natural course, there could have been a risk to mother and child.
He estimated the baby would be kept in hospital for three to four weeks while she gained weight and would then be able to go home.
Mrs Brown was expected to remain at the hospital for several days before being allowed home.
Consultant paediatrician Dr Laura Stewart was part of the team that helped deliver the child.
She said: "There were moist eyes all round. The baby is very cute.
"I think she looks like her father but her father says she looks more like her mother."
Baby Leo
Mr Brown, who is 50, and 37-year-old Sarah were married in a private ceremony in Fife, Scotland, in August last year.
They began dating in 1996 when Mrs Brown's PR firm, Hobsbawm and Macaulay Communications, was involved in arranging Labour Party events.
The birth comes less than two years after the Blairs produced the last Downing Street baby in May last year.
A spokesman at Downing Street said that Mr Blair and his wife Cherie, who are on holiday in Egypt, had sent their congratulations to the Browns.
He said: "The prime minister and Mrs Blair were absolutely delighted to hear the news of the birth of Gordon and Sarah's baby daughter and send their very warmest congratulations to all three."
Other politicians who sent their congratulations include Scottish Secretary Helen Liddell, shadow chancellor Michael Howard, Scotland's First Minister Jack McConnell, Scottish National Party leader John Swinney and Jim Wallace, leader of the Scottish Liberal Democrats.