The Duke and Duchess of Cambridge have thanked hospital staff for their "tremendous" care, as the world awaits the first glimpse of their baby son.
A spokesman said: "Mother, son and father are all doing well."
It's thought the family will leave hospital on Tuesday evening or on Wednesday morning.
Prince William was at St Mary's Hospital, in west London, for the birth at 16:24 BST on Monday, and stayed with his wife and son overnight.
'Absolutely beautiful' baby
In a statement, the couple said: "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."
At around 5.30pm on Tuesday William's father the Prince of Wales, arrived with his wife the Duchess of Cornwall at St Mary's Hospital to visit the young family.
Kate's parents, Carole and Michael Middleton, also visited the couple at the hospital on Tuesday afternoon.
As they left St Mary's Hospital, Kate's mum Carole Middleton said the baby was "absolutely beautiful" and the parents were "doing fabulously".
A gun salute has taken place in London to mark the birth of the Prince of Cambridge and Westminster Abbey's bells have been ringing all afternoon.
The royal baby was born weighing 8 lbs and 6 ounces in St Mary's Hospital in London and is third in the line to the throne.
The Prince of Cambridge, as the baby will be known, moves ahead of Prince Harry in line to the throne, because the law on royal succession has been changed.
Thousands of people went to Buckingham Palace when the news broke to see the announcement of the baby's birth on the easel on Monday evening.
Following the announcement, a statement from Kensington Palace said: "The Queen, The Duke of Edinburgh, The Prince of Wales, The Duchess of Cornwall, Prince Harry and members of both families have been informed and are delighted with the news."
The baby's name is yet to be announced but there's plenty of speculation about what it might be called.