Cuban-American Pedro Olivera Montane shouts anti-Castro slogans in the Little Havana section of Miami

The White House has asked Cubans on the island and exiles in Florida to remain where they are, amid speculation about the health of President Fidel Castro.

Spokesman Tony Snow said Wednesday it's not a time to attempt water crossings in either direction. He added officials have stressed the importance of eventually finding a way for safe and orderly travel.

A State Department spokesman, Sean McCormack, said the U.S. government stands ready to help the Cuban people if they show a desire to transition to democracy.

Cuban President Fidel Castro, speaks in Havana, September 9, 2002 (File photo)

Cuban state-run television reported Tuesday that President Castro was in stable condition after abdominal surgery.

Cuba's National Assembly president, Ricardo Alarcon, said the 79-year-old's final moment is "very far away."

Cuban media report that Raul Castro has assumed the presidency temporarily from his brother, who has not been seen since the surgery.

U.S. officials say this is the first time Fidel Castro has not been in charge since taking power in Cuba in 1959.

In southern Florida, Cuban exiles have celebrated the news of Mr. Castro's illness with spontaneous street parades and other festivities in Miami's Little Havana district.

Some information for this report was provided by AFP, AP and Reuters.