The news came as Kate, 30, was admitted today to King Edward VII Hospital in Central London with hyperemesis gravidarum, or acute morning sickness, which requires supplementary hydration and nutrients.
"As the pregnancy is in its very early stages, Her Royal Highness is expected to stay in hospital for several days and will require a period of rest thereafter," the statement said.
There was no confirmed information provided on when the baby is due. Several media reports, however, indicated Kate was approaching the 12th week of her pregnancy.
The palace said the Queen, the Duke of Edinburgh, the Prince of Wales, the Duchess of Cornwall, Prince Harry and members of both families are delighted with the news of the pregnancy.
"They will make wonderful parents," British Prime Minister David Cameron tweeted following the announcement.
In Canada, Prime Minister Stephen Harper tweeted his good wishes to the royal couple on behalf of himself and his wife: "Laureen and I congratulate the Duke and Duchess of Cambridge on their wonderful news #royalbaby."
Gov. Gen. David Johnston sent a congratulatory message from Peru, where he is travelling on a diplomatic mission to Central and South America. He noted that the royal couple had attracted large and appreciative crowds when they toured Canada last year. "With this happy news, we have once again a chance to show our admiration," he wrote.
White House press secretary Jay Carney expressed congratulations to William and Kate on behalf of U.S. President Barack Obama.
William and Kate were married at Westminster Abbey in April 2011 and speculation about a possible royal pregnancy has been media fodder ever since.
William is second in line to the throne behind his father, Prince Charles. The new baby will be third in line.
In 2011, Britain, Canada and the other countries that have the Queen as their head of state agreed to change the rules so that sons would not have precedence over daughters in the line of royal succession.
A spokesman for the British Cabinet Office confirmed to CBCNews.ca that if the child of Prince William and Kate is a girl, there will be no question that she could succeed to the throne.
"Even though the bill hasn't actually been introduced here yet, that doesn't matter. It's effectively law already," the spokesman said.
The British Cabinet Office hopes to introduce a bill soon.
The Queen already has two great-grandchildren, both from Peter Phillips, son of Princess Anne, and his Canadian wife Autumn.
Queen Elizabeth ascended to the throne on Feb. 6, 1952. At 86, she is exceeded only by Queen Victoria as the United Kingdom's longest-serving monarch. Victoria ruled for 63 years, seven months.