The couple, known as the Duke and Duchess of Cambridge, are "very pleased" and William's grandmother Queen Elizabeth II is "delighted", a spokesman said.
Kate, 32, is again suffering from hyperemesis gravidarum, a form of very acute morning sickness that caused her to be hospitalised when she was pregnant with George.
She is currently being treated by doctors at the couple's London home in Kensington Palace, the spokesman said.
The duchess was due to attend an official engagement in Oxford on Monday but cancelled because of her illness. William, also 32, will be attending the event as planned.
"Their Royal Highnesses The Duke and Duchess of Cambridge are very pleased to announce that The Duchess of Cambridge is expecting their second child," the palace announced.