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The baby is third in line to the throne
The Duke of Cambridge has said he and his wife are "still working on a name" for their son, who has spent his first night at home after leaving hospital.
The world got its first glimpse of the unnamed prince as the royal couple left St Mary's Hospital, west London.
Speaking to journalists on the steps of the hospital, William said they would choose a name "as soon as we can".
But royal fans may have to wait some time - in 1982 Prince William's name was only announced after a week.
George is the bookmakers' favourite for the name of the prince who is destined to be king one day. James and Alexander are among other names on a short price.
The prospects of him being named after his father has been virtually written off by the bookmakers, with both Ladbrokes and Paddy Power rating William as a 50/1 shot.
Asked by the BBC's royal correspondent Peter Hunt if the baby had already been named George, Prince William replied: "Wait and see, Peter, wait and see."
Prince William and Catherine appeared relaxed and smiled as they appeared with the new arrival on the steps of the hospital in front of a mass of photographers.
The duke told reporters: "He's got a good pair of lungs on him, that's for sure. He's a big boy, he's quite heavy."
He said: "It's the first time we have seen him really so we are having a proper chance to catch up."
William said he would remind his son "of his tardiness when he's a bit older".
He added: "I know how long you've all been standing here so hopefully the hospital and you guys can all go back to normal now and we can go and look after him."
A smiling duchess said: "It's such a special time. I think any parent will know what this feeling feels like."
The couple also revealed William had done the first nappy change.
The couple then went back inside the private Lindo Wing and placed their son in a car seat.
A few minutes later they came outside again and the duke drove his family home to Kensington Palace.
Earlier in the afternoon, the couple were visited at the hospital by the Prince of Wales and Duchess of Cornwall, and Kate's parents, Carole and Michael Middleton.
Prince Charles said the baby was "marvellous" while Mrs Middleton said he was "absolutely beautiful".
Speaking to reporters as she left the hospital, Mrs Middleton said: "We are so thrilled".
She added her first cuddle with her new grandson had been "amazing".
Prince William was at the hospital for the birth at 16:24 BST on Monday, and stayed with his wife and son, who weighed 8lb 6oz (3.8kg), overnight.
Kensington Palace said in a statement: "We would like to thank the staff at the Lindo Wing and the whole hospital for the tremendous care the three of us have received.
"We know it has been a very busy period for the hospital and we would like to thank everyone - staff, patients and visitors - for their understanding during this time."