Royal baby: Catherine spends second day in hospital NEW: UK bookmakers take bets on the likely name for the couple's first child William arrives at King Edward VII Hospital to visit his pregnant wife, Catherine Catherine suffers hyperemesis gravidarum, more severe than typical morning sickness The pregnancy is in "its very early stages," the palace says London (CNN) -- Britain's Catherine, Duchess of Cambridge, is spending a second day in a central London hospital Tuesday after being admitted with acute morning sickness. 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. Acute morning sickness: What is it? 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. 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. 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.