The Prince of Wales has said it is "a great joy" to have another grandchild, adding "I don't know how I am going to keep up with them".
Gun salutes across London have marked the arrival of the Duke and Duchess of Cambridge's third child.
The couple's second son was born on Monday at the Lindo Wing of St Mary's Hospital, London, weighing 8lb 7oz.
The new prince, whose name has yet to be revealed, has been receiving his first visitors at Kensington Palace.
His arrival was celebrated with traditional gun salutes from the Tower of London and Hyde Park, which are used to mark every birth of a prince or princess.
Westminster Abbey also marked the occasion with bell ringing - a tradition for significant royal occasions.
Catherine's sister Pippa was among the first visitors to Kensington Palace, where she met her nephew for the first time.
The Honourable Artillery Company fire a 62-gun salute at the Tower of London
In a statement released by Clarence House, Prince Charles said: "We are both [the Duchess of Cornwall and I] so pleased at the news.
"It is a great joy to have another grandchild, the only trouble is I don't know how I am going to keep up with them."
The palace said a name would be revealed in due course but for the time being a formal notice of the birth of the child has been placed on the forecourt of Buckingham Palace.
Arthur, James, Philip and Albert are among the bookies' favourites.
The Kings Troop Royal Horse Artillery fire a 41-gun salute from Hyde Park
The prince's siblings, Prince George and Princess Charlotte, visited their younger brother in hospital on Monday.
The duchess was admitted to the Lindo Wing at about 06:00 BST on Monday with the birth being announced shortly after 13:00 BST.
Crowds had gathered outside the hospital in Paddington, with some well-wishers having camped out for more than two weeks.
As Prince William left with his new son, he said the couple were very happy and joked he would have "thrice the worry now".
Before, younger royal males would have overtaken their older sisters.