The 35-year-old is again suffering from hyperemesis gravidarum, a condition that plagued her two previous pregnancies, and is being treated at home.
She has not yet reached the 12-week stage when women normally make the news public but made the announcement after being unable to attend an event in London on Monday afternoon, a spokeswoman said.
William, also 35, is second in line to the throne and the new baby will be fifth in line, pushing William's younger brother Harry down the order of succession.
"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.
Queen Elizabeth II, William's grandmother, and his father Prince Charles were said to be "delighted" at the news, while Harry said he was "very, very happy for them".
The news comes as William and Kate's eldest child, four-year-old Prince George, who is third in line to the throne, prepares to start school in London on Thursday.
The couple also have a daughter, two-year-old Princess Charlotte.
The palace declined to say when the new baby was due, but a spokeswoman confirmed to AFP that Kate "hasn't reached the 12-week stage yet".
Many women wait until they are 12 weeks pregnant to announce the news, due to the risk of miscarriage.
"As with her previous two pregnancies, the Duchess is suffering from hyperemesis gravidarum," the official palace statement said.
"Her royal highness will no longer carry out her planned engagement at the Hornsey Road Children's Centre in London today.
"The Duchess is being cared for at Kensington Palace."
Kate was only around a month pregnant with Charlotte when the news was announced, again because she had to cancel a public engagement.
Kensington Palace said in a statement: "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 Queen and members of both families are delighted with the news."
UK Prime Minister Theresa May tweeted her congratulations to the couple, calling it "fantastic news".
'Have more babies' -
Prime Minister Theresa May was quick to offer her congratulations, saying: "This is fantastic news."
Speaking at an event in Manchester, Harry said it was "great" that he was going to be an uncle again, and said of Kate: "I think she's okay."
Kate was hospitalised with hyperemesis gravidarum 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.
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.
There has long been speculation that the couple 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.
