Ernesto, which had strengthened into a hurricane for about 10 hours, weakened back into a tropical storm by late afternoon with top sustained winds of 60 mph, the National Hurricane Center said.
Still, the Miami-based hurricane center said the storm was expected to strengthen and potentially reclaim its hurricane status before reaching the southeastern coast of Cuba on Monday morning.
Ernesto became the first hurricane of the Atlantic season early Sunday with winds of 75 mph.
Bush's order came after the Monroe County Emergency Management office told tourists with immediate plans to travel to the Keys to postpone their trips and ordered those already in the island chain to leave. All travel trailers and recreational vehicles also were ordered off the islands immediately.
The National Hurricane Center in Miami said the storm could grow by Thursday into a hurricane as strong as Katrina, which struck New Orleans a year ago Tuesday.
"It's over nice warm Caribbean waters, and far enough off the coast of Haiti that it is still strengthening now," said Ron Goodman, a forecaster at the center.
New Orleans residents were told Saturday that the partially repaired levee system . Many residents – who lived through Katrina – are already preparing to evacuate the city if Ernesto heads toward Louisiana.
The storm was lashing Haiti on Sunday with heavy rain and sustained wind of 75 mph – raising concerns about deadly mudslides in the heavily deforested country.
The hurricane center said Haiti and the Dominican Republic could get up to 20 inches of rain in some places, which could cause life-threatening flash floods and mud slides.
"These people could be in great danger," said Adel Nazaire, a coordinator with Haiti's civil protection agency. "Flooding is the biggest concern because a lot of residents live along the rivers and the sea."
Ernesto was expected to cross west-central Cuba on Tuesday night before continuing into the Gulf of Mexico.
Cuba's six easternmost provinces have been warned to get ready — Hurricane Ernesto is heading directly toward them, CBS News producer Portia Siegelbaum. Special warning is out for residents of low-lying areas along the southeastern coast and in the mountains which are expected to be pelted with intense rain. Coastal flooding is expected along both the southeastern and northeastern coasts.
Cuban Civil Defense has begun evacuating people and livestock from the most vulnerable areas, Siegelbaum reports. Because of the poverty and lack of personal cars, evacuation is not left up to the individual. Evacuations here are a virtual military operation.
Ernesto was expected to bring rain and wind to southern Florida by early Tuesday, and the hurricane center encouraged people in southern Florida, the Florida Keys and Mexico's Yucatan Peninsula to monitor the storm. It was projected to strengthen off western Florida on Wednesday but the location of any U.S. landfall was unclear.
Jamaica's Prime Minister Portia Simpson Miller put the country's security forces on alert and said at a news conference Saturday that all the island's shelters were open.
"Ensure that the children are not left alone, and make it easier for rescue workers," she said.
Jamaica issued advisories by radio and television for residents in low-lying areas across the island to be prepared to evacuate if necessary.
At 11 a.m., the fifth named storm of the Atlantic hurricane season was centered about 115 miles southwest of Port-au-Prince, Haiti, and 205 miles southeast of Guantanamo, Cuba. Hurricane-force winds of at least 74 mph extended up to 15 miles from the storm's center.
The Royal Caribbean cruise line said it was diverting three ships scheduled to depart the United States on Sunday and Monday, sending them to alternative Caribbean ports to avoid the storm.
The hurricane center said the storm's 75 mph winds pushed it just above the threshold for a Category 1, the weakest category of hurricane. To reach Category 3, Katrina's strength, the winds would have to reach at least 111 mph.
"There will be probably be a restrengthening after it leaves the Cuban coast to a Category 2, and Wednesday night it will be west of Fort Myers as Category 3. That's the current thinking," Goodman said.
Heavy showers hit Kingston on Saturday afternoon, causing traffic jams as motorists tried to reach stores. People waited in long lines at supermarkets, filling grocery carts with canned goods, batteries and candles.
"It's nature and we can't stop it from taking its course," said taxi driver Patrick Wallace, 55, as he left a supermarket after stocking up on canned goods.