Forecasters said Fay, which earlier left at least five people dead in Haiti and the Dominican Republic, crossed central Cuba early Monday and was expected to pass near the Florida Keys that night.
A hurricane watch remained in effect for the capital, Havana, and eastward to Cuba's central Sancti Spiritus province.
A tropical storm watch was issued Monday for the northwestern Bahamas.
Cuban state media reported little damage or major flooding so far, but authorities in four provinces evacuated nearly 5,000 residents from low-lying communities and pulled fishing boats from the water. Officials also set up temporary shelters and food distribution centers.

In central Cienfuegos province, officials suspended traditional carnival celebrations. State media said authorities were ready to "protect" the 24,000 foreign tourists in the famous beach resort of Varadero, but provided no more details.
Winds damaged the roofs of some homes in little-populated areas and water accumulated on roads and highways.
The U.S. National Hurricane Center in Miami said Fay is expected to dump up to 8 inches of rain on Cuba, with 12 inches in isolated cases. It warned that much rain could produce flash floods and mudslides.
In the city of Niquero, near the southern coast and one of the hardest-hit areas, authorities converted a hotel into a shelter for evacuees.
"It's raining intensely, but the wind comes and goes," said a receptionist at the Hotel Niquero, who said he was not authorized to have his name appear in the foreign press.
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 Officials also suspended some ferry service on Isla de la Juventud, an island off Cuba's southern coast. In the southeastern province of Granma, a banana plantation sustained minor flooding and storm winds damaged some homes, state media reported.
At 5 a.m. EDT Monday, the storm's center was located over central Cuba and about 155 miles south-southeast of Key West and was moving toward the north-northwest near 12 mph. Maximum sustained wind speeds were near 50 mph. Forecasters expected the storm to begin moving to the north soon.
Florida has declared a state of emergency and authorities in the Florida Keys closed schools, opened shelters and urged visitors to leave. Residents and tourists, however, seemed in no hurry to evacuate.

Traffic leaving Key West and the Lower Keys on Sunday afternoon was light but steady as the sky darkened with storm clouds and the National Weather Service issued watches and warnings.
Fay, the sixth storm of the 2008 Atlantic season, was expected by forecasters to strengthen slowly to a hurricane.
[Associated
Press; By WILL WEISSERT]
Copyright 2008 The Associated
Press. All rights reserved. This material may not be published,
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