Prince William says he and the Duchess of Cambridge are "thrilled" they are expecting their second child.
Kensington Palace said the duchess was suffering from a severe sickness - as she did when she was pregnant with Prince George.
Speaking at a visit to Oxford University the duke said it had been a "tricky few days" but was "great news".
The couple's second baby will become fourth in line to the throne, behind 13-month-old Prince George.
Asked about the duchess, Prince William said she was "feeling okay" and that he was "going to go back and look after her now".
He told journalists the couple were "immensely thrilled", but said it was still "early days".
"It's important that we all focus on the big news, the big international and domestic things that are going on at the moment. That's what my thoughts are at the moment," he said.
A statement from the palace said the Queen and both families were delighted with news of the pregnancy.
Prince Harry said it was "very exciting news" and joked that he "can't wait to see my brother suffer more".
Speaking while visiting competitors ahead of the first Invictus Games, he said Catherine was "pretty poorly" but was "doing the best she can".
Prince William was speaking after he formally opened a £21m China study centre at Oxford University.
The duchess had been due to accompany him on the visit, but the palace said her sickness meant that she would no longer be attending and was being treated by doctors.
The duchess's first pregnancy was revealed when she was just a few weeks pregnant with Prince George after she was admitted to hospital suffering from severe sickness in December 2012.
The sickness - called hyperemesis gravidarum - is a condition that may require supplementary hydration, medication and nutrients.
It affects 3.5 per 1,000 pregnancies, causes severe vomiting and can lead to dehydration, weight loss and a build-up of toxins in the blood or urine, called ketosis.
Arriving at the new study centre at about 13:30 BST, the duke told a well-wisher who sympathised with Catherine's condition that she may be over the worst of the acute sickness in a "few weeks' time".
The duchess is scheduled to be among the guests at the opening ceremony of the Invictus Games in London on Wednesday evening along with Prince William, Prince Harry, the Prince of Wales and the Duchess of Cornwall.
On Thursday she is also due to attend a Drumhead service at Lee Valley Athletics Centre with William and Harry, and later with William to watch Invictus competitors taking part in matches and heats.
The duchess is due to make her first official solo overseas tour, to Malta for two days, from 20 September.
Kensington Palace said the duchess's attendance at future events would be decided on a "case-by-case" basis.
The baby announcement came after the couple's first child, Prince George, had his first birthday on 22 July.
Prime Minister David Cameron offered his congratulations to the Duke and Duchess of Cambridge in the House of Commons.
"We have heard this morning the Duke and Duchess of Cambridge are expecting their second baby.
"On behalf of the whole country, I'm sure the whole House will want to join me in congratulating them on this fantastic news and wishing them well in the months ahead," he said.
Labour leader Ed Miliband added his own good wishes, saying: "I join with the Prime Minister in congratulating the Duke and Duchess of Cambridge on their happy news.
"I too wish them well in the months ahead."
Earl Spencer, Prince William's uncle, told LBC radio station it was "fantastic" news.
"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," the earl, brother of the late Princess Diana, said.
Earlier, Deputy Prime Minister Nick Clegg said the pregnancy was "fantastic news".
Scottish First Minister Alex Salmond said: "This is very happy news for any couple, and on behalf of the people of Scotland, I am delighted to send our best wishes to the countess and our hearty congratulations to the royal couple."
The duchess is known as the Countess of Strathearn in Scotland.