The Duke and Duchess of Cambridge have emerged from St Mary's Hospital in west London with their new baby son.
Walking out to cheers from staff and well-wishers, Kate cradled her son and said it was "a special time".
Prince William said they were "still working on a name".
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.
A small crowd had gathered outside the palace to greet them.
Speaking to reporters outside the hospital earlier, the couple both said the experience was "very emotional".
William and Kate appeared relaxed and smiled as they appeared on the steps of the hospital shortly before 19:15 BST in front of a mass of photographers.
Kate handed their baby, who was wrapped in a white shawl, over to her husband and the couple walked forward to speak to reporters.
"He's got a good pair of lungs on him, that's for sure," William said.
"He's a big boy, he's quite heavy. We are still working on a name so we will have that as soon as we can."
He added: "It's the first time we have seen him really so we are having a proper chance to catch up."
A smiling duchess said: "It's such a special time. I think any parent will know what this feeling feels like."
Addressing the waiting crowd, Prince William said: "I'll remind him of his tardiness when he's a bit older.
"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."
They also revealed William had done the first nappy change.
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".
There has been no word yet on what the couple plan to name the prince and asked if she had been told or made any suggestions, Mrs Middleton said: "Absolutely not."
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."
The first public appearance of the royal baby attracted tweets from more than 18,000 people a minute, social network Twitter said.
But the figure was short of the peak of 25,300 tweets per minute on Monday night after his birth was announced, it said.
Meanwhile, the birth of the third in line to the throne was marked by gun salutes at Green Park and the Tower of London and the ringing of bells at Westminster Abbey.
And at Buckingham Palace, royal watchers and tourists queued in the rain to catch a glimpse of the bulletin - displayed on an easel - announcing the arrival of the royal baby.
The birth of the prince means the monarchy has three generations of heirs to the throne for the first time since 1901.