A statement from the Palace said both families are delighted.
It read: "Their Royal Highnesses The Duke and Duchess of Cambridge are very pleased to announce that The Duchess of Cambridge is expecting their second child.
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"The Queen and members of both families are delighted with the news.
"As with her first pregnancy, The Duchess is suffering from Hyperemesis Gravidarum. Her Royal Highness will no longer accompany The Duke of Cambridge on their planned engagement in Oxford today. The Duchess of Cambridge is being treated by doctors at Kensington Palace."
The new baby will be a brother or sister for Prince George
Her doctors are the surgeon gynaecologist to the Queen, Alan Farthing, and the surgeon gynaecologist to the royal household, Guy Thorpe-Beeston.
The Palace has said the pregnancy has not yet reached the 12-week point, but the announcement was made because the duchess was due at a series of public engagements in the coming weeks which are now in doubt.
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Prince George's sibling will be known as a Prince or Princess of Cambridge – just as he is HRH Prince George of Cambridge.
This follows a decree from the Queen in January 2013 that all children born to William and Kate should be styled Prince or Princess. Without that ruling, only the eldest son of the eldest son of the Prince of Wales would be entitled to the honour.
Prince Charles has been very vocal about his delight in becoming a grandfather and barely stopped beaming for months after George's birth.
Kate is suffering from severe morning sickness again
For Kate, the timing is perfect because her family is well settled in Apartment 1A, their home at Kensington Palace, which has been totally refurbished to her taste.
Work has also been carried out at Anmer Hall, which is thought to have been allocated to the Cambridges by the Queen as a country retreat.
The listed ten-bedroom Georgian property is two miles from the Queen's residence on the Sandringham Estate.
Out of concerns for privacy, the drive was re-routed and more trees were planted to shield the property. There were also plans drawn up to expand accommodation.