The royal family are "delighted" by the news of the baby, who would be the third in line to the British throne currently occupied by Queen Elizabeth II.
"Their Royal Highnesses The Duke and Duchess of Cambridge are very pleased to announce that The Duchess of Cambridge is expecting a baby," St James's Palace said in a statement.
"The Queen, The Duke of Edinburgh, The Prince of Wales, The Duchess of Cornwall and Prince Harry and members of both families are delighted with the news."
William, the son of heir to the throne Prince Charles and the late Princess Diana, married the former Kate Middleton in April 2011 and there has been intense speculation about when the couple, who are both 30, would have a baby.
The statement added: "The Duchess was admitted this afternoon to King Edward VII Hospital in Central London with Hyperemesis Gravidarum (severe morning sickness).
"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."
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