"The Duchess of Cambridge is continuing to feel better," a St. James's Palace spokesman told CNN Tuesday afternoon.
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 the hospital for several days, the palace said Monday.
The palace said her illness is hyperemesis gravidarum, which involves nausea and vomiting more severe than the typical morning sickness many women suffer during early pregnancy.
"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.
Deputy Prime Minister Nick Clegg said Tuesday that all 16 countries that recognize the British monarch as their head of state have formally agreed to the change and British lawmakers will change the rules as soon as possible.
Clegg also said that the change in succession will allow someone in line to the throne to marry a Catholic -- but not to be a Catholic. The Church of England is the nation's official church. It split from the Roman Catholic Church in the 16th century.
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."
British Labour Party leader Ed Miliband tweeted: "Fantastic news for Kate, William and the country. A royal baby is something the whole nation will celebrate."
Catherine is an art history graduate and the eldest child of Michael and Carole Middleton, self-made millionaires who run a party-supplies company.
Tabloid speculation about a pregnancy increased in September when Catherine substituted water for wine during a state dinner toast in Singapore.
The buzz reached a boil two weeks ago when Australia's New Idea magazine quoted close friend Jessica Hay saying her former schoolmate, Catherine, would soon announce "wonderful, happy news."
Life & Style magazine in the United States published its own "exclusive" with the same Hay quotes of a December announcement, which she said she had on "the highest authority."
A family history website did the math last summer and predicted Catherine would be with child by November, according to a Time.com report. Genesreunited.co.uk calculated that "the average period between a royal wedding taking place and the couple's child being born is 851 days." Based on that, it predicted a pregnancy in November and a birth on August 27, 2013.
CNN's Matthew Fitzgerald and Jamie Hamilton in London contributed to this report.