Kate, 31, gave birth to her first child in the private Lindo Wing at St Mary's Hospital in Paddington, London at 4.24pm today (Monday, 22 July). The baby boy weighed 8lbs 6oz.
The child, who will be known as the Prince of Cambridge, is third-in-line to the throne after Prince Charles and Prince William.
A name has not been announced yet.
Kate and William chose not to learn their baby's gender ahead of the birth, meaning today's arrival was a surprise for them and their families.
The birth was announced via press release after Buckingham Palace dramatically switched their plans at the last minute. Just after 8pm this evening, the palace revealed they'd issue a press release before the official birth announcement was taken to Buckingham Palace.
No reason was given for the switch but some commentators suggested it was to minimise news of the royal birth being leaked before the official easel announcement.
Traditionally a royal birth is announced in the following way: the medical staff sign an official birth announcement that is taken by a royal aide - via police escort - to Buckingham Palace where it is attached to an easel for the public to see.
This did happen today, but not until after the press release had been issued.
The news of the birth was also revealed in a series of tweets posted to the Clarence House account: "Her Royal Highness The Duchess of Cambridge was safely delivered of a son at 4.24pm. The baby weighs 8lbs 6oz.
"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 & families have been told and are delighted."
"Her Royal Highness and her child are both doing well and will remain in hospital overnight."
Her Royal Highness The Duchess of Cambridge was safely delivered of a son at 4.24pm— Clarence House (@ClarenceHouse) July 22, 2013
Her Royal Highness and her child are both doing well and will remain in hospital overnight.— Clarence House (@ClarenceHouse) July 22, 2013
News of the birth was greeted by loud cheers at Buckingham Palace and outside St. Mary's Hospital, with further cheers when the royal couple's press secretary emerged from the hospital to pass the birth notice to a driver, who took it to Buckingham Palace.
Media had been camping outside St Mary's Hospital for the past two weeks awaiting news of Kate's labour; a different scene to years ago. Prince William was actually the first royal to be born at a hospital. His father, Prince Charles, was born at Buckingham Palace.
The royal birth was overseen by the Queen's former gynaecologist Marcus Setchell.
Kate arrived at the private Lindo Wing at around 6am this morning, accompanied by Prince William. A release from the palace at 7.30am said the Duchess was in the "early stages of labour". A further statement said Kate's labour was "progressing as normal".
It's believed the world will get its first glimpse at the royal arrival when Kate and William leave the hospital and pose for a traditional picture on the front steps.
It will also be our first glimpse of the Duchess since her last public engagement in June, when she attended the Trooping the Colour ceremony at Buckingham Palace to celebrate the Queen's birthday.
William is expected to take two weeks' paid paternity leave from his job as an RAF search and rescue pilot, while St. James' Palace has said Kate's maternity leave is a private matter.
The palace added before the birth: "[This is a] very personal matter for the duke and duchess, but they also know it's a time to celebrate and many will want to share in their joy."
Kate and William, who married in a televised ceremony in April 2011, announced their pregnancy in December 2012, but the happy news was marred slightly by Kate's hospitalisation with severe morning sickness.
The Duchess spent several days at King Edward VII hospital in London being treated for hyperemesis gravidarum.
Kate Middleton's pregnancy in pictures: