The Duke of Cambridge said he is "thrilled" at the prospect of becoming a father for the second time.
Reflecting on "a tricky few days" for the Duchess, he said the announcement came earlier than planned due to another bad case of morning sickness.
The Queen has expressed her delight at the news - and Prince Harry said it was all very exciting.
ITV News Royal Correspondent Tim Ewart reports:
Prince William's uncle Earl Spencer has welcomed the "fantastic" news that he and wife Kate are expecting their second child.
"It is just such happy-making news," he told LBC Radio. "I have never known a period of news where things have looked so dark and bleak around the world and to have this pop up is just fantastic.
"I think that's really what it's about, it's just making the nation a happier place and that is what today's news does."
Prince Harry has joked about the news he is likely to be another step away from being King following the Duchess of Cambridge's second pregnancy.
Asked about being pushed one further down the line, the Royal laughed and said: "Great!"
Prince Harry has welcomed the news of the Duchess of Cambridge's pregnancy as "very exciting" - joking that he cannot wait to see his brother "suffer more".
He said Kate was coping with her illness and expressed his wish that the couple would be able to enjoy some privacy during her pregnancy.
The Duke of Cambridge says his wife is "feeling OK" after being treated for hyperemesis gravidarum after falling pregnant with her second child - adding that he was going home to "look after her".
Speaking at Oxford University today, he said: "It's been a tricky few days - week or so - but obviously we're basically thrilled, it's great news, and early days.
"We're hoping things settle down and she feels a bit better."
"It's important that we all focus on the big international and domestic news that is going on at the moment," the Duke said.
The Duke of Cambridge today said he was "thrilled" after it was announced the Duchess of Cambridge is expecting their second child, adding it had been "a difficult few days".
The Duchess of Cambridge, who it was announced today is expecting her second child, may be over the worst of her acute morning sickness in a "few weeks time", the Duke of Cambridge said today.
Royal historian Dr Kate Williams has said that it must be "very hard for William and Kate" after they were forced to announce the Duchess' pregnancy early, meaning "they don't have that early period of privacy".
As with her last pregnancy the Duchess of Cambridge has suffered from hyperemesis gravidarum (severe morning sickness).
This has meant she has been prevented from carrying on her royal duties, forcing the palace to confirm both pregnancies earlier than most parents might go public.
Expectant parents usually wait until after the first trimester.
Royal historian Kate Williams has said that being the 'spare heir' in the royal family can be a difficult position.
Dr Williams described Princess Margaret as: "the example of someone who didn't enjoy being number two."
The Duke of Cambridge arrived alone to formally open The Dickson Poon University of Oxford China Centre Building.
He had been due to be accompanied by his wife but she was unable to attend due to severe morning sickness.
The Duke of Cambridge accepted a small fluffy toy and a bunch of flowers on his arrival - possibly the first gift for the new royal baby.