The Duchess of Cambridge is responding well to treatment for severe pregnancy sickness after spending a second day in hospital, St James's Palace said.
The Duchess was "continuing to feel better" and was "immensely grateful" for the messages of support she has received since being admitted on Monday.
Her improvement was reflected in the Duke of Cambridge's expression as he left the King Edward VII Hospital in London wearing a smile after spending almost six-and-a- half hours at his wife's bedside.
The Duchess is still expected to spend several more days in hospital, where she is thought to have been put on a drip to prevent dehydration after being diagnosed with hyperemesis gravidarum, which causes repeated vomiting.
A St James's Palace spokesperson 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."
The Duchess is being treated by Marcus Setchell, the former gynaecologist to the Queen. His successor, Alan Farthing, was also at the hospital yesterday, where he is thought to have been consulted on the Duchess's treatment.
The Duke, 30, arrived in a green Land Rover Discovery at 11.30am wearing an open-necked shirt, jumper, jeans and trainers.
He walked straight inside, without glancing at the journalists and television crews from around the world on the other side of the street.
When he left shortly after 5.45pm there was a noticeable difference in his mood, and he made no attempt to hide his apparent relief at his wife's improvement. The Duchess did not receive any other visitors, though her parents and siblings are expected to travel today from their home in Bucklebury, Berkshire.
The British Government announced yesterday (Tuesday) that it had received final consent from all the Commonwealth realms to change the laws of succession. The Succession to the Crown Bill will be debated in Parliament "at the earliest opportunity" to ensure that the couple's first born child has its place in the succession enshrined in law, regardless of its sex.
The law will also remove the bar on anyone in line to the throne marrying a Roman Catholic, although the ban on Catholics becoming sovereign will remain because of the monarch's position as head of the Church of England.
Bookmakers cut the odds on the Duchess having twins from 50/1 to 8/1 after The Daily Telegraph reported yesterday that her medical condition was associated with a threefold increase in the chance of a multiple birth. Betting on the child being a girl may also increase after it was suggested that the Duchess's illness and lifestyle factors make her statistically more likely to have a daughter.
The odds on triplets, meanwhile have come down from 1,000-1 to 100-1.
Victoria is now the most heavily backed name for the new baby, coming down from 20-1 to 8-1, with Elizabeth, Diana and Charles among other favourites. Surprisingly, punters have also placed money on the baby being called Wayne or Chardonnay, both at 500-1.
The bookmaker also said it had taken "hundreds of pounds" on the baby having ginger hair, like its uncle, Prince Harry, at odds of 6-1.