The baby was born at 4:24 p.m., and weighs 8 lbs, 6 oz.
The announcement said:
"The Duke of Cambridge was present for the birth.
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.
Her Royal Highness and her child are both doing well and will remain in hospital overnight."
She was admitted to the private Lindo wing of St. Mary's Hospital Monday morning in the early stages of labor, a statement from Kensington Palace said.
Her husband Prince William was with her, according to the statement.
They traveled to St. Mary's from Kensington Palace by car, not by ambulance. The west London hospital is where William himself was born in 1982.
The duchess is thought to be about one week past her due date, according to the BBC's royal correspondent, Peter Hunt.
Officials would not comment on whether her labor would have to be induced, he said.
The royal baby will be third in line to the British throne, after his grandfather, Prince Charles, and father, Prince William.
The child's royal title will be the Prince or Princess of Cambridge, Buckingham Palace announced earlier this month.