The Duchess of Cambridge's parents said the new Prince of Cambridge is "absolutely beautiful."
The Duke and Duchess are "both doing really well. We're so thrilled," said Carole Middleton on the steps of the Lindo Wing as she left the hospital.
She said she had "absolutely not" given a suggestion for the child's name.
The Duke of Cambridge has said he "we could not be happier" at the birth of his son.
A Kensington Palace spokesman said Prince William would be staying with the Duchess and their son at the hospital overnight.
Sharon Thomas sent this report from Windsor.
The Duchess of Cambridge's parents, Carole and Michael Middleton, have arrived at the Lindo Wing of St Mary's Hospital.
They emerged smiling from a black taxi and waved at the assembled media before entering the building.
The Duchess of Cambridge's parents, Carole and Michael Middleton, have arrived at the Lindo Wing of St Mary's Hospital to visit their newborn grandson.
The names of royal babies are historically not announced straight away, leaving the nation playing a guessing game for several days.
When Princess Beatrice was born in 1988, it was two weeks before her name was known.
When William was born in 1982, his parents the Prince and Princess of Wales waited seven days before deciding upon and announcing his name.
The Prince of Wales's name, however, remained a mystery for an entire month and was only declared ahead of his christening in the Music Room of Buckingham Palace in December 1948.
– Charles Kidd, editor of Debrett's Peerage and Baronetage
I think the Royal Family quite like to do things slowly.
It's quite dignified to have a bit of breathing space between the birth and the announcement of the name. It's a measured approach.
Out of courtesy, the Queen has to be told what has been decided.
The bookmakers favourite name for the royal baby might be George but social historian Dr Judith Rowbotham, of Nottingham Trent University, told the Daily Telegraph that it is an unlikely option.
Despite the name (George) being a bookies’ favourite, royal advisers are likely to point out that the 19th century Prince George of Cambridge was not exactly a model that (Kate and William) would wish the new prince to emulate.
Though a successful military commander, that Prince George of Cambridge was not popular with his cousin Queen Victoria, especially after he married an actress, Sarah Fairbrother, in 1847. She was, at that time, the mother of four children by three different fathers.
Gun salutes are sounding across London to mark the birth of the royal baby.
The King's Troop Royal Horse Artillery and the Honourable Artillery Company are both carrying out the ceremonial salutes in honour of the new addition to the Royal Family.
Gun salutes are fired for the birth of every prince or princess, no matter where their place is within the line of succession, the Ministry of Defence said.
The King's Troop Royal Horse Artillery, wearing full dress uniform, rode past Buckingham Palace to Green Park where they are staging a 41 royal gun salute.
The Honourable Artillery Company (HAC), the City of London's Army Reserve Regiment, are firing a 62 royal gun salute from Gun Wharf at the Tower of London.
The London Gazette has published an official notification of the birth of the Prince of Cambridge, who was born yesterday at 4.24pm.