Gordon and Sarah Brown wept tears of joy yesterday at the birth of their son.
The Chancellor's wife gave birth at Edinburgh's New Royal Infirmary. Mother and baby were said to be fine.
Prime Minister Tony Blair was among those who expressed delight at the news.
He said: "This is wonderful news. Cherie and I are absolutely delighted.
"I have passed our very best wishes to Gordon and Sarah. We are delighted that mother and baby are doing so well."
The couple's reaction to the birth was revealed in a statement outside the hospital given by the doctor who delivered their son.
Dr Rhona Hughes, consultant obstetrician at the New Royal Infirmary, said: "I'm delighted to be able to tell you that Sarah Brown gave birth to a healthy baby boy this morning at 9.28am here at the Simpsons Centre for Reproductive Health, in Edinburgh.
"He weighed 8lb 1oz. Mother and baby are both doing very well.
"Sarah and Gordon are absolutely delighted. It was a very emotional moment and they both wept tears of joy."
The baby is the couple's second child. Their daughter, Jennifer Jane, died in January last year weeks after being born prematurely.
The Chancellor's elder brother, John, said: "The whole family are delighted at the birth of Sarah and Gordon's baby. He is a beautiful boy."
He refused to speculate on what the boy would be called, including the suggestion that he may be named John.
"That's a name which is common in our family, but the name of the baby is for Gordon and Sarah, and no doubt they will say that during the course of the next two or three days," he told reporters.
Mr Brown said he did not know when the couple would be leaving the hospital.
He also said there had been lots of good wishes from Mr Brown's constituency, across the political spectrum, as well as from various parts of the country and even abroad.
Conservative Party leader Iain Duncan Smith and Charles Kennedy, leader of the Liberal Democrats, were among political figures who sent their congratulations to the Chancellor and his wife.
Among visitors to the hospital yesterday were Mrs Brown's mother, Pauline MacAulay, and stepfather Patrick Vaughan. They left the maternity unit without commenting.
Harry Potter author JK Rowling, who has a home in Edinburgh, also visited the couple. She told reporters: "He's a beautiful, beautiful baby."
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