William, 35, is second in line to the throne and the new baby will be fifth in line, bumping William’s younger brother Harry down the order of succession.
The news comes at the start of the week that the royal couple’s eldest child, four-year-old Prince George, begins school in London.
They also have a daughter, two-year-old Princess Charlotte.
“Their royal highnesses the Duke and Duchess of Cambridge are very pleased to announce that the Duchess of Cambridge is expecting their third child,” the palace said in a statement.
There was no immediate indication when the baby was due or whether it is a boy or a girl.
Queen Elizabeth II, William’s grandmother, was said to be “delighted” at the news, as were members of both families.
Prime Minister Theresa May was quick to offer her congratulations, saying in a message on Twitter: “This is fantastic news.”
The palace confirmed that “as with her previous two pregnancies, the duchess is suffering from Hyperemesis Gravidarum”, an acute form of morning sickness.
Kate, 35, was hospitalised with the condition during her first pregnancy in 2012, while it also forced her to cancel a trip to Malta when she was pregnant with Charlotte in 2014.
“Her royal highness will no longer carry out her planned engagement at the Hornsey Road Children’s Centre in London today,” the palace said.
“The Duchess is being cared for at Kensington Palace.”
William and Kate moved back to London from their rural home in eastern England this summer as they take over more engagements from the ageing senior royals.
The 91-year-old queen has reduced her public events in recent years and Prince Philip, her 96-year-old husband, officially retired in August.
William gave up his job as an air ambulance pilot in July and relocated his family from Anmer Hall in Norfolk to their apartment at Kensington Palace.
Prince George, the third in line to the throne, is due to begin school at the private Thomas’s Battersea school in south London on Thursday.
There has long been speculation that William and Kate would like a third child.
On a royal tour of Poland in July, the duchess — who herself is one of three children — joked about having another after being given a cuddly toy designed to soothe tiny babies.
Saying thank you for the present, she turned to William and joked: “We will just have to have more babies.”
Hyperemesis gravidarum causes excessive nausea and vomiting and affects around one in every 100 pregnant women, according to the state-run National Health Service (NHS).
Some women report being sick up to 50 times a day, and while most common in early pregnancy, symptoms can continue throughout the nine months. DM
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