The Duchess of Cambridge is “continuing to feel better” and she and the Duke are immensely grateful for the good wishes they have received, St James's Palace has said.
Kate spent a second night in a private hospital where she is being treated for a severe form of morning sickness after revealing her pregnancy.
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.”
William spent more than six hours with his wife at the Edward VII hospital in central London yesterday and left at 5.45pm looking relaxed, smiling at the waiting reporters and photographers.
Kate was admitted to hospital on Monday after developing the condition that can leave patients feeling tired, dizzy and suffering from headaches — all signs of dehydration.
But it now appears she is responding to treatment and is likely to be having less severe bouts of morning sickness.
The Queen's former gynaecologist Marcus Setchell, who delivered the Countess of Wessex's two children, is treating her, according to reports.
Kate's diary of engagements for this week have been cancelled to allow for the extensive rest she will need when she is discharged.
The prolonged vomiting could continue for much of her pregnancy — which is still below 12 weeks — and she may need to take anti-sickness medication for months to come.
The sex of the baby is not known but whatever name the royal couple choose for their first child, it is likely to set a trend.
Elizabeth has been by far the best backed name and has been cut from 25/1 to 10/1 joint favourite alongside Victoria and George by bookmaker William Hill.
News that Kate's condition could suggest she is carrying twins has also prompted the firm to cut the price of the royal having two babies from 33/1 to 7/1.
Paddy Power have slashed their odds on twins from 50/1 to 8/1 after a rush of bets and Victoria is the most heavily backed name, with odds falling from 20/1 to 8/1.
Law will ensure newborn can be sovereign... even if it’s a girl
By Gavin Cordon
Legislation to ensure the Duke and Duchess of Cambridge's baby becomes monarch — whether it's a boy or a girl — is to be introduced as “rapidly” as possible, Deputy PM Nick Clegg has said.
The 15 nations where the Queen is head of state have given their final consent for the Government to press ahead with the landmark Bill to end the centuries old discrimination against royal women in line to the throne.
Mr Clegg said ministers would now introduce the Succession To The Crown Bill in the House of Commons at the “earliest opportunity” available in the parliamentary timetable.
The legislation will end the principle of male primogeniture, so that the first child of the Duke and Duchess of Cambridge will succeed to the throne, regardless of whether the baby is a girl or a boy.
The legislation will also end the bar on anyone in the line of succession marrying a Catholic.
“This is a historic moment for our country and our monarchy,” Mr Clegg said.
“People across the realms of the Commonwealth will be celebrating the news that the Duke and Duchess of Cambridge are expecting their first child. We can now also all celebrate that whether this baby will be a boy or a girl, they will have an equal claim to the throne.
“It's a wonderful coincidence that the final confirmation from the other realms arrived on the very day that the Duke and Duchess of Cambridge made their announcement.”
The new rules will apply to any child born in the line of succession after the Commonwealth Heads of Government meeting in Perth, Australia, in October 2011 when the 16 countries which have the British monarch as their head of state agreed in principle to the change.
Since then, the New Zealand government has been co-ordinating the formal consents from each realm, with assurances that they will be able to complete the necessary measures before the British legislation comes into force.
It will involve amendments to some key constitutional documents, such as the Bill of Rights and Coronation Oath Act of 1688, the 1701 Act of Settlement and Act of Union with Scotland.
“It will write down in law what we agreed back in 2011 — that if the Duke and Duchess of Cambridge have a baby girl, she can one day be our Queen even if she later has younger brothers,” added Mr Clegg.
Royal pregnancy gives birth to media frenzy across the globe
By James Edgar and Rob Williams
THE announcement that the Duchess of Cambridge is pregnant with a future king or queen sparked a media frenzy as journalists descended from all corners of the globe on the London hospital where Kate is being treated for severe morning sickness.
Her admission to the King Edward VII private hospital provided a focal point for international news coverage, and the world's press camped out around the clock to catch a glimpse of the Duke of Cambridge and other members of the royal or Middleton families as they visited her bedside.
Film crews and photographers from Australia, the United States, China, Japan, Brazil and all over Europe are likely to stay until Kate's condition improves enough for her to leave hospital.
Around 20 television cameras and dozens of photographers with long lenses waited behind barriers over the road from the hospital's entrance, wrapped in winter clothes and standing on ladders to get a better view.
Its arrival in the world may still be many months away but the Royal baby has already acquired an impressive number of spoof Twitter accounts.
Within minutes of the pregnancy announcement, the micro-blogging site was awash with new accounts. Parody accounts were set up including UnbornRoyal, HRHBaby and IamRoyalBaby.
The first tweet from RoyalFoetus read: “HELLO WORLD: ALL SHALL KNEEL BEFORE ME.”
A commemorative mug celebrating the happy news has gone into production — less than 24 hours after the official announcement.
High-end 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. But the manufacturer confirmed that sample products had been drawn up and the first batch was heading for the kiln.
The half-pint mug bears red and blue hearts and has the words “A Royal Baby in 2013” on one side and “Hooray for Will & Kate” on the other. The commemorative pieces will cost £19.95.
The company, based in Lichfield Street, Hanley, said the souvenirs would continue a two-century-old tradition of manufacturing royal memorabilia.
Many Americans appear to be as enthralled as Britons over the impending royal baby.
“I'm always looking for any news of William and Kate,” said 19-year-old Stacy McFacken, a clerk at a grocery store in Mentor, Ohio. “There's nothing like this in the States,” she said. “It's just like all the fairy tales we read about as kids. We all want to be Kate.”