The baby girl was delivered at 8.34am today weighing 8lbs 3oz, Kensington Palace confirmed.
Her Royal Highness The Duchess of Cambridge was safely delivered of a daughter at 8.34am.— Kensington Palace (@KensingtonRoyal) May 2, 2015
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.
Speculation over when the Duchess would leave the hospital was prompted when her hairdresser, Amanda Cook Tucker, arrived at the Lindo Wing.
Kate is now expected to leave by 7pm, according to Kensington Palace.
Prince William left briefly to go home to pick up Prince George and bring him to visit his new baby sister.
The youngster has now returned to Kensington Palace for "bath and bed".
The huge numbers of journalists, photographers and broadcasters who had assembled outside the Lindo Wing were caught a little off guard by the speed of the announcement of the birth of the new Princess.
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."
An easel was also placed in the Forecourt of Buckingham Palace to announce the birth .
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."
All the party leaders tweeted their congratulations but the strangest reaction was from Nigel Farage, who said he was happy the baby's weight was announced in imperial measurements.
He said: "Imperial measurements, proper measurements - eight pounds three ounces.
"Not some horrid kilogramme measurement, I'm very, very pleased. Perhaps that tells us where the royal family really stand."
He added the birth of the royal baby was an "exciting and joyous event for millions of patriotic British people".
"I strongly urge all Ukip candidates to take at least an hour out of campaigning in order to toast the new arrival. I shall certainly be setting an example in that regard. After all, some things are worth interrupting politics for," he said.
The new baby is the second grandchild for heir to the throne Prince Charles.
The baby pushes Prince Harry down to fifth in line to the throne and she could one day be head of state of 16 countries (which is probably a lot of responsibility to be reckoning with on the first day of your life...)
The baby girl is a princess, but only because of the Queen's intervention.
Under past rules limiting titles within the royal family, the girl would have been Lady Mountbatten-Windsor.
But the Queen issued a Letters Patent under the Great Seal of the Realm in December 2012 when Kate was around three months pregnant with George, declaring ''all the children of the eldest son of the Prince of Wales should have and enjoy the style, title and attribute of royal highness with the titular dignity of Prince or Princess prefixed to their Christian names or with such other titles of honour''.
When William is king, the new Princess might have the honorary style the Princess Royal - which is currently used by Princess Anne. It is customarily given by the sovereign to his or her eldest daughter.
Following a hospital spell for severe morning sickness, Kate continued with her royal engagements throughout the pregnancy, finishing with a visit to the Stephen Lawrence Centre in late March.