The Duke and Duchess have one son, George, and one daughter, Charlotte, aged four and two.
"The queen and members of both families are delighted with the news", the statement said.
This is published unedited from the PTI feed.
The child will become the fifth in line to the British throne and, if the child is a boy, will be the first in the immediate royal family to be affected by changes to the line in succession.
Canadian economy grows above expectations in Q2
GDP has averaged annual growth rates of just 2.2 percent in this recovery, which is now the third longest in US history. And I happen to be one that thinks we can go much higher than three percent".
Kensington Palace made the announcement in a statement on Monday.
Their choice of Thomas's Battersea in south London indicated that the Cambridges were settling into their Kensington Palace apartment, having moved recently from their Norfolk home Anmer Hall.
Bookies are already offering odds on what Princess Kate and Prince William will name their third child - and Alice has come out on top, so far.
A change in law after William and Kate Wednesday gave women the same rights of accession to the throne as men.
Kate is the eldest of three siblings, and reportedly had a very happy childhood.
Prince Charles, Queen Elizabeth II's oldest son, is now No. 1 in the succession line.