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 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."