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 says.
Media frenzy over UK royal baby news
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Media frenzy over UK royal baby news
Media from all over the world camp outside the London hospital where Kate Middleton, the Duchess of Cambridge, is being treated for acute morning sickness.
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 yesterday.
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."
Couple decided to go public
The royal couple made the decision to go public with news of the pregnancy and scrambled to tell family and friends before the story broke on social media, say reports in London.
Even the Queen, 86, who receives daily briefings on international affairs of the upmost secrecy, was kept in the dark until Monday when the whole world also learnt the Cambridges are expecting their first child.
Officials were forced to go public because of fears Catherine's hospitalisation would fuel concerns over her health, The Guardian newspaper reported.
However British broadsheet The Daily Telegraph told a different story: "royal aides said the decision to go public 'was very much driven by the Duke and Duchess'.
"It's a very hard thing to go public at such an early stage, but they wanted to be open with people as much as possible."
So read the first of more than five pages in Tuesday's edition dedicated to the baby, its possible name, its path to the throne, the effect he or she will have on British tourism, and of course, details of the morning sickness which has landed Catherine in hospital.
Although pregnant for less than 12 weeks, the common point at which expectant mothers share their news, Catherine and William feared the information would leak and be repeated on social networking sites such as Twitter.
There would have been no way of avoiding suspicion once the Duchess was forced to cancel upcoming official engagements, as she did on Monday.
First baby of online age
From spoof Twitter accounts to feverish speculation about names, the internet has gone into a frenzy over the unborn child of William and Kate as the first royal baby of the online age.
News of the former Kate Middleton's pregnancy - announced by the royal family on Twitter - met with an explosion of posts on social networks, from joyous congratulations to those pleading for the media coverage to end already.
It is perhaps of little comfort to Catherine, in hospital for a second day on Tuesday with severe morning sickness, that within minutes of the announcement her baby already had a slew of spoof accounts "live-tweeting from the royal womb".
"CURRENT STATUS: DARK IN HERE, WILL UPDATE," tweeted RoyalFoetus, which has 6000 followers.
The rival RoyalFetus, which has 9000 followers, added: "I may not have bones yet, but I'm already more important than everyone reading this. #royalbaby #sorry".
The hashtag #royalbaby instantly rocketed to the top of Twitter's "trending topics" list on the announcement.
Legislation to end discrimination
The UK government has received final consent from all the Commonwealth realms to press ahead with legislation ending discrimination against women in the line of succession to the British throne, Deputy Prime Minister Nick Clegg says.
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 Roman Catholic.
"This is a historic moment for our country and our monarchy. People across the realms of the Commonwealth will be celebrating the news that the Duke and Duchess of Cambridge are expecting their first child," Mr Clegg said.
"We can also all celebrate that whether the baby is 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."