"The wedding will take place in the Spring or Summer of 2011, in London. Further details about the wedding day will be announced in due course.
"Prince William and Miss Middleton became engaged in October during a private holiday in Kenya.
"Prince William has informed The Queen and other close members of his family. Prince William has also sought the permission of Miss Middleton's father.
"Following the marriage, the couple will live in north Wales, where Prince William will continue to serve with the Royal Air Force."
A Buckingham Palace spokeswoman said: "Both the Queen and the Duke of Edinburgh are absolutely delighted for them both."
Meanwhile the Countess of Wessex said: "I think it's absolutely thrilling. We couldn't be more excited."
Speaking of the royal family's reaction she said: "I think everyone's absolutely thrilled and we wish them all the love and luck in the world."
Mr Cameron was informed of the royal engagement during the Cabinet's meeting this morning and said the news was met with a "great cheer" and with "a great banging of tables".
He said that he had passed on his congratulations to Prince William during a telephone call this morning.
He described the Prince as "very excited" and said he had told him that the Government would do all it could to ensure the occasion was a great success.
He also hinted that while the wedding will be a public occasion it was still the case of "two young people who love each other" who deserved privacy.
Deputy Prime Minister Nick Clegg and his wife Miriam also sent their best wishes.
"I am thrilled to hear the fantastic news of Prince William and Kate Middleton's engagement," Mr Clegg said.
"Miriam and I would like to send our very best wishes to the happy couple ahead of their big day."
And Labour leader Ed Miliband added: "I'm delighted for Prince William and Kate Middleton and send them my very best for their wedding.
"The whole country will be wishing them every happiness."
Catherine Elizabeth Middleton, 28, is the eldest child of Michael and Carole Middleton, a businessman and former air stewardess.
She met Prince William while they were both students at St Andrews University and have been dating for eight years.
A middle class family Middletons live in Bucklebury, a small village near Newbury, Berkshire, and run a mail-order business called Party Pieces which sells toys and party paraphernalia.
Miss Middleton previously worked in the accessories department of Jigsaw but left the job to join her parents' firm. She has two younger siblings, Pippa and James, who also work for the family business.
She was privately educated at Marlborough College and first came to public attention in 2002 when she was identified as one of three friends with whom William was to share a house while studying at St Andrews.
She is credited with persuading the Prince to continue his studies when he considered quitting as he struggled to adjust to university life during his first year.
