“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.
According to details, Princess Catherine was taken to the Lindo Wing of St Mary’s Hospital in the early hours of Saturday morning.
The Duke of Cambridge was present for the birth of the baby, who weighs 8lbs 3oz (3.7kg).
The duchess had earlier been admitted to the Lindo Wing at St Mary’s Hospital where Prince George was born in July 2013.
The birth of second child to the Duchess attracting huge attention around the world.
The interest in the Royal Baby on social media in Pakistan could be gauged from the fact that the hashtag #RoyalBaby was trending at number 1 on Twitter.
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.
Another royal supporter, John Loughrey, said: “I’m so happy. I can’t tell you. It’s a lovely feeling. I’m feeling very happy for the royal family.”
Royal fan Kathy Martin donned a pink party hat and started dancing on the spot. “It’s the happiest news, just what we’ve been waiting for,” she said.
Kensington Palace said Kate was admitted to the hospital accompanied by her husband at around 6.00 am in the early stages of labour.
British Prime Minister David Cameron sent his best wishes in a tweet.
“The whole country will wish her well,” he wrote shortly after she was admitted.
Within minutes of the news that Kate had gone into labour, the news was trending on Twitter with the hashtag #royalbaby and more and more curious onlookers could be seen arriving at the private clinic in the city centre.
“I heard it on the radio and I got up and came straight here!” Michelle Lassiter, a 55-year-old charity worker who travelled from her home in Guildford near London with a fold-up chair, told AFP.
Guy Thorpe-Beeston, surgeon-gynaecologist to the royal household and a specialist in high-risk pregnancies, led the delivery team.
Kensington Palace had announced Kate’s pregnancy in September and she struggled with severe morning sickness as she did in her first pregnancy.
William has told supporters the second child will be a “game-changer” after the first “life-changer”.
– Gun salutes in celebration –
William and Kate are planning to spend the first few days after the birth at Kensington Palace.
They will then travel to Anmer Hall, a secluded 10-bedroom country mansion on Queen Elizabeth’s privately-owned Sandringham estate in Norfolk, eastern England, where they have set up home.
Cannons will be fired across London to celebrate the birth — 62 shots from the Tower of London and 41 from Green Park — and the British flag will be flown from government buildings.
The baby’s name may not be revealed for several days.
William’s name was not announced for a week, while the world had to wait one month after Charles was born.
George was named two days after his birth.
The princess will be christened as a member of the Church of England, wearing a replica of the intricate lace and satin gown made for queen Victoria’s eldest daughter in 1841.
Experts say the royal birth could inject tens of millions of pounds into the British economy, with a baby princess particularly lucrative because she could become a fashion trendsetter.
Congratulations on Twitter
Congratulations to the Duke and Duchess of Cambridge on the birth of their baby girl. I’m absolutely delighted for them.
— David Cameron (@David_Cameron) May 2, 2015
Congratulations to the Duke and Duchess on the birth of their daughter. Wishing them lots of joy and happiness – and hopefully some sleep! — Ed Miliband (@Ed_Miliband) May 2, 2015
Congratulations to the Duke & Duchess of Cambridge on the birth of their new baby. Miriam & I wish them all the best as their family grows
— Nick Clegg (@nick_clegg) May 2, 2015
— Royal Navy (@RoyalNavy) May 2, 2015