The baby was delivered at 16:24 BST at St Mary’s Hospital in Paddington, west London, weighing 8lb 6oz.
The palace said in a statement that the duchess and the baby were “doing well” and would stay in hospital overnight.
The news has been displayed on an ornate easel in the forecourt of Buckingham Palace in line with tradition.
A bulletin – signed by the Queen’s gynaecologist Marcus Setchell, who led the medical team that delivered the baby – was taken by a royal aide from St Mary’s to the palace under police escort.
The press release said the Duke of Cambridge was present for the birth.
“The Queen, The Duke of Edinburgh, The Prince of Wales, The Duchess of Cornwall, Prince Harry and members of both families have been informed and are delighted with the news,” it said.
Prime Minister David Cameron said on Twitter: “I’m delighted for the Duke and Duchess now their son has been born. The whole country will celebrate. They’ll make wonderful parents.”
And Labour leader Ed Miliband, also writing on Twitter, said: “Many congratulations to the Duke and Duchess of Cambridge. I wish them and their son all happiness and good health.”
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