"Mother, son and father are all doing well this morning," a Kensington Palace Spokesman 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," a statement from the Duke and Duchess read.
"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 new family are expected to leave the hospital either after 6pm on Tuesday evening or on Wednesday morning.
The world's media remained camped outside the private wing of St Mary's Hospital, Paddington where Kate gave birth to the little Prince weighing 8lb 6oz at 4:24pm on Monday afternoon.
Photographers, cameramen, reporters and well-wishers alike are eager to get the first glimpse and photo of the new family.
As well as seeing the first photo of the baby boy and his proud parents, the public are eager to know what name the Duke and Duchess of Cambridge will choose for the third in line to the throne.
The statement added that there is currently "no news on names."
It seems likely that members of the Middleton family will turn up to visit Kate, William and their infant son.
However, it has been suggested that members of the royal family will not visit the hospital. First-time grandfather Prince Charles and his wife Camilla, Duchess of Cornwall have engagements in Yorkshire and new uncle Prince Harry is at RAF Wattisham with his Apache squadron.
The Queen, who received a phone call from grandson William on Monday afternoon to tell her of the happy news, is at Buckingham Palace.