Catherine, wife of Prince William, smiled and waved as she left hospital on Monday after giving birth to a boy who is now fifth in line to the British throne.
The baby was born at 11.01am and weighed 3.8 kg. William was present for the birth at St Mary's Hospital in west London where their other two children, George and Charlotte, were also born.
Kate, wearing a red dress, smiled at hundreds of well-wishers and assembled journalists as she left the hospital clutching her baby just seven hours after the birth. William later put the boy, strapped into a car seat, into a waiting car.
"Thrice the worry now," William told reporters, holding up three fingers with a smile, as he got into a Land Rover to take his wife and newborn back to Kensington Palace. "We didn't keep you waiting too long this time."
"Very happy, very excited, thank you," he said. When asked about a name for the boy, William said: "You'll find out sooner or later." Bookmakers have made Albert, Arthur, Jack and Fred the favoured names for a boy.
Queen Elizabeth, Prince Charles, the baby's grandfather, and other members of the royal family and Kate's family have been informed and "are delighted with the news", Kensington Palace said.
The baby is the queen's sixth great-grandchild and is behind his grandfather and heir Prince Charles, father William, and siblings George and Charlotte in line to the throne.
A 2013 change to the law means that for the first time in British history, a new prince will not supplant his older sister in the order of succession. Previously younger males would take precedence over older female siblings.
William's younger brother Harry, who marries US actress Meghan Markle next month, falls to sixth.
The official announcement of the birth was made by placing a notice on an easel in the forecourt of the queen's London home, Buckingham Palace, a tradition that has been in place since it became the sovereign's official residence in 1837.
William and Kate met as students at St Andrews University in Scotland. They wed at Westminster Abbey in April 2011.
The couple now live at Kensington Palace and last year William gave up his day job as an air ambulance pilot to concentrate full-time on royal duties on behalf of his grandmother, having faced some criticism from British newspapers who dubbed him "workshy Wills".
He has been determined to play a hands-on role in bringing up his children while also protecting their privacy, an issue close to his heart after the death of his mother, Princess Diana, in a 1997 Paris car crash as her limousine sped away from pursuing paparazzi.
"I think he takes on a lot more engagements than he used to," said royal biographer Claudia Joseph.
"But compared to members of the general public who have full-time jobs, he still has some time off to look after the children, to give them their baths, to put them to bed and I think he'll continue to do that."
