Tropical Storm Ernesto has dumped more than three inches (7.5cm) of rain as it travels across eastern Cuba.
Ernesto is expected to reach Florida on Tuesday or Wednesday
Cuban authorities moved at least 500,000 people from eastern provinces, but no damage has been reported.
Winds have weakened to 40mph (65km/h) but meteorologists warned those in Ernesto's path not to relax their guard as it heads for the Bahamas.
They say it could regain hurricane status before reaching Florida on Tuesday or Wednesday.
Ernesto battered the Dominican Republic and Haiti, where it killed one person, and Florida Governor Jeb Bush has declared a state of emergency.
Nasa cancelled Tuesday's planned space shuttle launch from Cape Canaveral in Florida and was deciding whether to move the orbiter indoors because of the approaching storm.
The centre of the storm was close to Holguin, eastern Cuba, at 1400 EDT (1800 GMT) on Monday, travelling north-west at about 10mph (16km/h) , said the National Hurricane Center (NHC) in Miami.
More than three inches of rain fell on the hills above Cuba's eastern city of Guantanamo, and the centre warned that parts of central and eastern Cuba and the island of Hispaniola could expect between six to 12 inches of rain.
The Cuban authorities made extensive preparations for Ernesto, moving at least 500,000 residents in the east, transferring cattle to higher ground and rescheduling baseball games.
Ernesto briefly became this year's first Atlantic hurricane on Sunday, but was downgraded to tropical storm status as it reached southern Haiti.
The NHC cautioned that a hurricane watch was still in force for the north-western Bahama islands, and much of coastal Florida - warning Ernesto could gather strength and reach winds of about 85mph (137km/h) as it approaches Florida.
Tourists have been ordered to leave the Florida Keys island chain south of the US state, and schools closed.
"I don't want anyone to overly focus on the downgrading... It has a good chance to regain hurricane status," said Max Mayfield, director of the NHC.
One person died when the storm lashed Vache island off Haiti's southern coast, local officials said.
A tropical storm becomes a hurricane when its sustained winds reach speeds of 75mph (120km/h) and above.
Ernesto comes a year after a strong Category 3 hurricane devastated New Orleans.
More than 1,000 people died and hundreds of thousands were left homeless when Hurricane Katrina flooded the city and surrounding Gulf Coast areas one year ago on Tuesday.
The city is still struggling to recover from the effects of Katrina, but New Orleans' emergency services say they are prepared for any new storms:
"We have a solid plan. All we need to do is watch the storm and the timing," City homeland security chief Terry Ebbert said.
Last year's Atlantic-Caribbean hurricane season was the worst on record, producing 15 hurricanes.