Kensington Palace announced moments ago: "The Duchess of Cambridge has given birth to a baby boy"
The baby was delivered at 16:24 BST at St Mary's Hospital in Paddington, west London, weighing 8lb 6oz.
Duchess and the baby were "doing well" and would stay in hospital overnight.
An ornate easel in the forecourt of Buckingham Palace is displaying the news in line with tradition.
A bulletin - signed by the Queen's gynaecologist Marcus Setchell - was taken by a royal aide from St Mary's to the palace under police escort.
The document said: "Her Royal Highness, the Duchess of Cambridge was safely delivered of a son at 4.24pm today.
"Her Royal Highness and her child are both doing well."
The Kensington Palace 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 stated.
The Prince of Wales said he and the Duchess of Cornwall were "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," he added.
"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."
Prime Minister David Cameron Tweeted: "I'm delighted for the Duke and Duchess now their son has been born. The whole country will celebrate. They'll make wonderful parents."
Deputy Prime Minister Nick Clegg Tweeted: "Congratulations to The Duke and Duchess of Cambridge. Good news to make the whole country smile."
And Labour leader Ed Miliband, also Tweeted: "Many congratulations to the Duke and Duchess of Cambridge. I wish them and their son all happiness and good health."
Scottish First Minister Alex Salmond said: "I am sure that people across Scotland will be absolutely thrilled to hear the news of the birth of a baby boy to the Royal couple and will want to join me in wishing the proud parents many congratulations."