The Duke of Cambridge has said his wife has experienced a "tricky few days" with very acute morning sickness – but they are thrilled at her pregnancy.
Speaking during a visit to Oxford University to open its new £21 million China Centre, the Duke said: "She's feeling OK, thanks. It's been quite a tricky few days – a week or so – but obviously we're thrilled, it's great news, and early days.
"We're hoping things settle down and she feels a bit better."
Kate had to pull out of the visit to open the China Centre with William as she is suffering from hyperemesis gravidarum, as she did during her first pregnancy.
On the advice of medical staff she remained at Kensington Palace where she is being treated by doctors.
It is understood that Kate's pregnancy has not passed the 12-week stage – a similar position to when she was struck by the illness when pregnant with Prince George.
And there is already speculation that the royal couple's second child could be due in April.
The news was greeted by the Queen and members of both William and Kate's families with joy – they are "delighted" at the news, Kensington Palace said.
And during William's visit to the China Centre, he was inundated with congratulations from staff and students.
The Duke repeatedly said "thank you" to the supportive words as he greeted crowds during a brief walkabout as he made the short journey through St Hugh's College to tour the new building.
William suggested that his wife could be back carrying out official duties in a few weeks when one well-wisher sympathised with Kate's condition and said they wished she could have attended.
The Duke replied: "I know, she wishes she could be here – a few weeks' time."
For the Duchess to cancel her attendance, her illness, which is thought to have started a few days ago, is likely to have caused some concern but was well managed.
Like her first pregnancy the onset of the condition has meant news of her second baby has probably been announced earlier than the Cambridges would have planned. William's speech at the centre was focused on the subject of the visit, but he did mention his wife's absence.
"I'm sorry that Catherine could not join me today. She is sorry to miss it, and she hopes to come back in the future to meet you all for herself," he said.
The pregnancy announcement comes around six weeks after Prince George celebrated his first birthday.
The couple's second child will become fourth in line to the throne, shifting Prince Harry further down the line of succession.
Kate's condition will throw into doubt whether she will be able to attend her first solo overseas tour, set to take place in Malta later this month.
The Duchess' first pregnancy was revealed when she was around eight weeks pregnant after she was admitted to hospital suffering from very acute morning sickness in December 2012.
Hyperemesis gravidarum, which 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.
A Kensington Palace spokesman said the Duchess's attendance at future events would be decided on a "case by case" basis.
Kate is scheduled to be among the guests at the opening ceremony of the Invictus Games in London on Wednesday evening, along with William, Harry and the Prince of Wales and Duchess of Cornwall.
She is due to make her first official solo overseas tour in two weeks' time, from September 20-21.
A spokesman said: "It's far too early to talk about Malta and Invictus."