LONDON (CBS) -- The world's media is gathered outside a British hospital where doctors are treating the Duchess of Cambridge for severe morning sickness. Kate's royal pregnancy is causing lots of excitement and even producing some bets.

Prince William was back on hospital duty this morning after spending a late night at his wife's bedside. The father-to-be is keeping an eye on Kate as doctors treat the Duchess for severe morning sickness.

While Kate recovers there is growing speculation about the couple's future heir to the throne. Will the royal baby be a boy or a girl? Some even wonder if Kate's expecting twins.

The Duchess is not yet 12 weeks pregnant, but betting shops in Britain are already taking wagers on baby names. Rupert Adams with William Hill Betting Shop says, "I think they are very traditional people, so if you look at the betting, there is George, Victoria, Elizabeth."

Outside the hospital, reporters from around the world are camped out. They're waiting to get the first shot of Kate since palace officials announced she's pregnant. Royal author Robert Hardman says, "This was kept very quiet wasn't known in Buck or Downing Street until hours or minutes before it was announced to the world."

Kate was last seen publicly four days ago playing field hockey in heels at an event at her former school. The Duchess may need to spend several days at the hospital. Doctors are still getting fluids into her system. They say her extreme nausea is treatable but for some expectant moms it can last until the baby is born.

The hospital released a statement today that says the Duchess is continuing to feel better and that she and her husband are grateful for the good wishes they've received.