The Duchess of Cambridge is pregnant with her second child.
Kensington Palace said that – as when the duchess was pregnant with Prince George – she was suffering from very acute morning sickness and was being treated by doctors at the palace.
Prince William and Catherine’s second baby will become fourth in line to the throne, behind older brother Prince George, who is 13 months old.
The palace said the Queen and both families were delighted with the news.
Prime Minister David Cameron, Labour leader Ed Miliband and the Archbishop of Canterbury were among those to offer their congratulations to the couple.
Comment by Peter Hunt, BBC News royal correspondent
For a second time, William and Kate have been forced to announce a pregnancy before the duchess passed the significant 12-week milestone.
For a second time it’s because Kate is suffering from an acute form of morning sickness – though this time she’s being treated behind palace walls and not at a private hospital with representatives of the world’s media gathered outside.
The pregnancy has generated international excitement and will continue to do so but this impending birth will lack the constitutional significance of Prince George’s arrival.
As things stand, the as yet unborn girl or boy isn’t destined to be a monarch. She or he will occupy the same position as Prince Harry once did. His mother, Diana, used to call him “the back up”.
Monday’s announcement has performed one unintended but useful role for the Windsors. Headlines about the Queen’s view of Scottish independence will be replaced by extensive coverage of a royal birth next year.
The duchess’s first pregnancy was revealed when she was just a few weeks pregnant with Prince George after she was admitted to hospital suffering from severe morning sickness in December 2012.
The sickness – called hyperemesis gravidarum – is a condition that may require supplementary hydration, medication and nutrients.
It affects 3.5 per 1,000 pregnancies, causes severe vomiting and can lead to dehydration, weight loss and a build-up of toxins in the blood or urine, called ketosis.
Catherine had been due to join Prince William in formally opening a £21m China study centre at Oxford University but the palace said her morning sickness meant that she would no longer be attending.
The Duke of Cambridge arrived at the new study centre at about 13:30 BST, where he was presented with a bouquet of flowers by a young girl.
He told a well-wisher who sympathised with Catherine’s condition that she may be over the worst of the acute morning sickness in a “few weeks’ time”.
Kensington Palace said the duchess’s attendance at future events would be decided on a “case-by-case” basis.
She is scheduled to be among the guests at the opening ceremony of the Invictus Games in London on Wednesday evening along with Prince William, Prince Harry and the Prince of Wales and Duchess of Cornwall.
On Thursday she is due to attend a Drumhead service at Lee Valley Athletics Centre with William and Harry, and later with William to watch Invictus competitors taking part in matches and heats.
The duchess is due to make her first official solo overseas tour, to Malta for two days, from 20 September.
The baby announcement came after the couple’s first child, Prince George, had his first birthday on 22 July.
Prime Minister David Cameron said: “Many congratulations to the Duke and Duchess of Cambridge. I’m delighted by the happy news that they’re expecting another baby.”
Mr Miliband said: “Fantastic to hear that Prince George will soon be a big brother. Congratulations to the Duke and Duchess of Cambridge on their happy news.”
Deputy Prime Minister Nick Clegg said the pregnancy was “fantastic news”.
Scottish First Minister Alex Salmond said: “This is very happy news for any couple, and on behalf of the people of Scotland, I am delighted to send our best wishes to the countess and our hearty congratulations to the royal couple.”
The duchess is known as the Countess of Strathearn in Scotland.