Her husband, Prince William, arrived Tuesday morning to visit her at King Edward VII Hospital, after spending much of the day at her bedside Monday.
The news that he and Catherine are expecting their first child after 19 months of marriage was announced by the palace Monday as she was admitted for treatment, and followed months of tabloid speculation.
The duchess is likely to remain in hospital for several days, the palace said Monday.
The palace said her illness is hyperemesis gravidarum, which involves nausea and vomiting more severe than typical morning sickness.
“As the pregnancy is in its very early stages, Her Royal Highness is expected to stay in hospital for several days and will require a period of rest thereafter,” the statement said.
Catherine, 30, is not yet 12 weeks pregnant, Clarence House told CNN, so the palace is not announcing a due date for the child.
William and Catherine’s child will be next in line to the British throne after William, regardless of whether it is a boy or a girl, the British Cabinet Office said Monday.
Planned changes to the law of succession that end the tradition of a boy jumping over an elder sister are already de facto in effect, the Cabinet Office said.
William and Catherine were married at Westminster Abbey on April 29, 2011, in London’s biggest royal wedding in three decades.
The baby would be the first grandchild of the heir to the British throne, Prince Charles, and the late Diana, Princess of Wales.
British bookmaker William Hill said it would be paying out “a small five-figure sum” to people who had placed bets on William and Catherine having a baby in 2013.
Wagers are now being taken on the likely name, gender and hair color of their first-born, the bookie said in a written statement.
It has Frances and John as the favorites for the baby’s name, at 9-to-1, with Charles, Victoria and George at 10-to-1.
“We fully expect that the betting public will get involved in speculating what the baby will be called,” said William Hill spokesman Joe Crilly.
Fellow bookmaker Ladbrokes has Elizabeth — the name of the baby’s great-grandmother — as an 8-to-1 favorite, with Frances, John, Charles and George at 10-to-1.
The news of Kate’s pregnancy immediately became a top trending topic worldwide on Twitter.
British Prime Minister David Cameron was quick to tweet his joy: “I’m delighted by the news that the Duke & Duchess of Cambridge are expecting a baby. They will make wonderful parents.”