Kate is spending a second night in a private hospital where she is being treated for a severe form of morning sickness after revealing her pregnancy on Monday.
William spent most of the day at the bedside of his wife, who is likely to be on a drip so she can receive fluids intravenously to combat the effects of dehydration caused by the condition, known as hyperemesis gravidarum.
Her illness could indicate she is having twins as mothers carrying two babies have a greater chance of developing the severe morning sickness.
But while there is concern for the royal couple, there is excitement across the UK and beyond, with messages of support sent from leading figures both at home and abroad.
A St James's Palace spokesman said: "The Duchess of Cambridge is continuing to feel better.
"She and the Duke are immensely grateful for the good wishes they have received.
"She will remain in hospital at present and will continue to be treated for hyperemesis gravidarum."
Her particular illness could indicate she is having twins as mothers carrying two babies have a greater chance of developing the severe morning sickness.
The world's media remain camped outside the private King Edward VII Hospital waiting for any news about the future queen and her unborn child.
So far, Victoria and George have been the best backed names during the first 24 hours since the announcement was made and they are both offered at 10/1.
News that Kate's condition may indicate she is carrying twins has prompted bookmaker William Hill to cut the price of the royal couple having two babies from 33/1 to 25/1 following a number of bets on Monday night.
Meanwhile, a commemorative mug celebrating the Duke and Duchess of Cambridge's happy news has gone into production – less than 24 hours after the announcement.
Pottery company Emma Bridgewater, which hand-makes all its products at its Victorian factory in Stoke-on-Trent, has yet to confirm a release date for the items out of respect for the royal couple.
But the manufacturer confirmed that sample products had already been drawn up and the first batch was heading for the kiln.