Prince George and Princess Charlotte arrived earlier in the day to meet their baby brother, but were not present when their parents took pictures outside the hospital.
Kensington Palace said the baby was born at 11am, after Kate was taken by car to the Lindo Wing just before 6am.
Bookmakers have reported a surge in bets on the name Arthur for the royal baby, who is fifth in line to the throne.
William was present for the birth this morning, and both Kate and the child were 'doing well', according to a palace spokesman. Senior members of the royal family were all said to be 'delighted with the news'.
Kensington Palace said: 'Her Royal Highness The Duchess of Cambridge will be leaving St. Mary's Hospital this evening. The Duke and Duchess of Cambridge and their son will travel home to Kensington Palace.'
A statement added: 'Prince George and Princess Charlotte have now returned to Kensington Palace. Their Royal Highnesses would like to thank all staff at the hospital for the care and treatment they have received.'