The Duchess of Cambridge has given birth to a baby son, weighing 8lb 60z, Kensington Palace has confirmed.
The royal baby was born at 4.24pm, but the announcement was not made until 8.30pm.
William, the Duke of Cambridge, issued a brief statement saying: "We could not be happier."
Earlier Kensington Palace 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.
The birth was also announced on an easel outside Buckingham Palace
"Her Royal Highness and her child are both doing well and will remain in hospital overnight."
A Buckingham Palace spokesman said: "The Queen and Duke of Edinburgh are delighted at the news."
The Prince of Wales described his joy at the birth of his first grandchild, saying how the arrival of a baby son was "an incredibly special moment for William and Catherine".
The heir to the throne said he and his wife the Duchess of Cornwall were "overjoyed" at the baby's arrival and "so thrilled" for William and Kate.
He reflected on the special status grandparents enjoy and said he was looking forward to meeting his new grandson soon.
The Prince said: "Both my wife and I are overjoyed at the arrival of my first grandchild. It is an incredibly special moment for William and Catherine and we are so thrilled for them on the birth of their baby boy.
"Grandparenthood is a unique moment in anyone's life, as countless kind people have told me in recent months, so I am enormously proud and happy to be a grandfather for the first time and we are eagerly looking forward to seeing the baby in the near future."
David Cameron tweeted that the "whole country would celebrate" the birth, while Nick Clegg said it was news to "make the whole country smile".
I'm delighted for the Duke and Duchess now their son has been born. The whole country will celebrate. They'll make wonderful parents.
— David Cameron (@David_Cameron) July 22, 2013
Congratulations to The Duke and Duchess of Cambridge. Good news to make the whole country smile
— Nick Clegg (@nick_clegg) July 22, 2013
Labour leader Ed Miliband said he wished the news parents and their son "happiness and good health".
Sky News reported that Prince William personally called The Queen, his father Prince Charles, his brother Prince Harry and the Middleton family to pass on the news.
Soon after the palace announcement was made, a notice giving details about the baby - the third-in-line to the throne - left St Mary's Hospital in London by car for Buckingham Palace.
It has been signed by the Queen's former gynaecologist Marcus Setchell, who led the medical team that delivered Kate's baby.
The new royal baby will be the Queen's third great-grandchild and is destined to be crowned monarch.
He will be the 43rd sovereign since William the Conqueror if, as expected, it follows reigns by the Prince of Wales and then William.
The Duchess was admitted to St Mary's Hospital in the early hours of Monday morning, arriving by car with Prince William.
After the press release was issued, the more traditional announcement was made at Buckingham Palace when a footman walked out of Buckingham Palace with the typewritten notice at 8.50pm.
The crowds outside the gates cheered as the footman, accompanied by another Palace official, attached the notice to the easel.
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