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 said on Tuesday evening.
Kate is spending a second night in a private hospital where she is being treated for hyperemesis gravidarum, 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."
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.
William spent more than six hours with his wife at the Edward VII hospital in central London and left at 5.45pm looking relaxed, smiling at the waiting reporters and photographers.
The royal couple are likely to be cheered by the development that legislation to ensure their baby becomes monarch - whether it is a boy or a girl - is to be introduced as "rapidly" as possible, according to deputy prime minister, Nick Clegg.
Mr Clegg said: "The Government will soon introduce the Succession to the Crown Bill which will make our old-fashioned rules fit for the 21st century.
"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."