The new baby was born at 8.34am weighing 8lbs 3oz and William was at his wife’s side during the delivery.
Kate was a number of days overdue and had been taken to the Lindo wing of St Mary’s Hospital at 6am from her nearby home Kensington Palace.
Kensington Palace said in a statement: “Her Royal Highness The Duchess of Cambridge was safely delivered of a daughter at 8.34am. The baby weighs 8lbs 3oz.
“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.”
The Duchess of Cambridge’s daughter is fourth in line to the throne, the Queen’s fifth great-grandchild and a spare to the heir - and Prince George’s sister.
The Prince of Wales will be overjoyed as he made no secret he wanted his second grandchild to be a girl.
Before the birth William and Kate said they did not know the sex of their baby, so the arrival of a daughter will be a surprise to them.
Kate’s daughter was delivered far quicker then her first child - with the baby being born less than three hours after she was admitted to hospital at 6am in the early stages of labour.
George took more than 10 hours to be born but it appears his sister might have taken less then half that time.
The Duchess is being looked after by consultant obstetrician Guy Thorpe-Beeston, who is the surgeon-gynaecologist to the household.
He was joined in the delivery room by Alan Farthing, the Queen’s surgeon-gynaecologist.
The huge numbers of journalists, photographers and broadcasters who had just assembled outside the Lindo Wing were caught a little off guard by the speed of the announcement of the birth.
A ripple of excitement went through the royal fans who had been camped outside the private maternity wing for days.
Town crier Tony Appleton - president of the Guild of International Millennium Town Criers - added a sense of drama to the occasion.
Standing on the steps of Lindo Wing and in a booming voice, he told the large crowds beginning to gather: “We welcome with humble duty the second-born of their Royal Highnesses the Duke and Duchess of Cambridge.
“The Princess is fourth in line to the throne.”
The political leaders took a moment out from General Election campaigning to congratulate the Cambridges on the birth of their daughter via Twitter.
David Cameron tweeted: “I’m absolutely delighted for them”. Labour leader Ed Miliband said: “Wishing them lots of joy and happiness - and hopefully some sleep!”.
Nick Clegg, of the Liberal Democrats, tweeted: “Miriam & I wish them all the best as their family grows.”
Scotland’s First Minister Nicola Sturgeon issued a brief statement sending the couple her “warmest congratulations”.