Kate, 31, gave birth to a baby boy weighing 8lbs 6oz at 4.24pm this afternoon (Monday, 22 July).
In a statement this evening, Prince Charles 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."
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."
Kate delivered the third-in-line to the throne in the private Lindo Wing at St. Mary's Hospital in London, while the world's media waited outside.
The boy, who will be known as the Prince of Cambridge, has not been named yet.
Prince Charles, meanwhile, spent today on an official engagement in York, where he joked with photographers they would probably know about the royal birth before he did.
The birth was announced via press release after Buckingham Palace dramatically switched their plans at the last minute. Just after 8pm this evening, the palace revealed they'd issue a press release before the official birth announcement was taken to Buckingham Palace.
After the press release, an aide was seen bringing the official notice - signed by medics and declaring the baby's sex and time of birth - to Buckingham Palace, where it is currently being displayed on an easel.
The news of the birth was also revealed in a series of tweets posted to the Clarence House account: "Her Royal Highness The Duchess of Cambridge was safely delivered of a son at 4.24pm. The baby weighs 8lbs 6oz.
"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 & families have been told and are delighted."
"Her Royal Highness and her child are both doing well and will remain in hospital overnight."
Kate arrived at the private Lindo Wing at around 6am this morning, accompanied by Prince William. A release from the palace at 7.30am said the Duchess was in the "early stages of labour". A further statement said Kate's labour was "progressing as normal".
It's believed the world will get its first glimpse at the royal arrival when Kate and William leave the hospital and pose for a traditional picture on the front steps.
Kate Middleton's pregnancy in pictures: