
"Her Royal Highness The Duchess of Cambridge was safely delivered of a daughter at 8.34am," the royal residence of Kensington Palace said in a statement.
Mother and baby "are both doing well," it said.
Prince William was present for the birth and all senior members of the royal family have been told the news, which was first announced in a tweet and then by the traditional colourfully-dressed town crier on the steps of the hospital.
The new Princess of Cambridge, who weighed in at 8 pounds 3 ounces (3.7 kilos), comes after big brother George in the line of succession, below Queen Elizabeth II's eldest son Prince Charles and his eldest son William.
The names favoured by bookmakers are Alice and Charlotte.
Media teams from around the world rushed to the private maternity wing of St Mary's hospital in the early morning, joining a group of royal superfans who have remained camped outside for more than a week.
There were cheers in the crowd after the birth, with supporters singing: "It's a girl! It's a girl!"
"I'm so pleased," Margaret Tyler, 71, who wore a jacket in the colours of the British Union Jack flag and a badge in honour of William's late mother Diana.
"William wanted a girl and Diana wanted a girl. I think it's just nice for George to have a little sister," she said.