Chancellor Gordon Brown said he "just loves being a dad" as he revealed his new son is to be called James.
Gordon Brown says he now has 'two great boys'
The 8lb boy, who was born at Edinburgh Royal Infirmary, will have the second name Fraser, which was chosen by his brother John, who is nearly three.
A beaming Mr Brown, said: "I just love being a dad, it's great fun - there's nothing more important."
Mr Brown, wearing a light blue shirt, said: "All of us are really happy with this wonderful addition to our family."
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The chancellor, who made the announcement to waiting reporters outside the hospital, said his wife Sarah was "extremely well".
"Our baby son is James and his other name is Fraser. He is very strong and he's very healthy," he said.
"I think he's going to be a pretty good footballer and he's got John's blue eyes."
Mr Brown, 55, who is the hot favourite to succeed Tony Blair as prime minister, said the name James was a family name.
He recalled that he had said they were the happiest family in Britain when John was born.
He then added: "We are a larger and happier - and I believe - we're a very grateful family today.
"We have got two great, marvellous boys."
The couple's first child, Jennifer Jane, who Mr Brown has said was an "inspiration", died aged 10 days after being born prematurely in 2002.
The chancellor said he had received congratulations from Tony Blair, political colleagues from all sides in the UK - and spoken to ex-South African President Nelson Mandela, who was celebrating his 88th birthday on Tuesday.