Ernesto became the Atlantic season's first hurricane on Sunday morning after being upgraded briefly from a storm and is heading straight towards the Kennedy Space Centre, disrupting Nasa's launch plans for Atlantis.
Nasa has announced plans to move the shuttle indoors to shelter it.
Space agency officials, under a tight timetable to complete work on the International Space Station, said launch attempts are unlikely until late next month or October.
No material damage
So far, Ernesto has caused little material damage, but a woman died as she was swept away by floods on a Haitian island.
But forecasters said that the storm is likely to strengthen again to a dangerous hurricane once it moves over the warm waters of the Gulf of Mexico.
"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 US National Hurricane Centre in Miami.
On Tuesday morning Ernesto was over open water just off Cuba's northern coast, about 370 kilometres southeast of Key West, Florida, and about 380 kilometres south-southeast of Miami.
Moving toward the west-northwest at near 22 kilometres an hour, with maximum sustained winds near 65 kilometres an hour, the centre of Ernesto was expected to be near the Florida Keys or southeast Florida by Tuesday evening.
Thousands were evacuated in Cuba, where the communist government regularly undertakes mass evacuations before major storms to minimise loss of life. There were no reports of major damage, but state television showed flooding in some eastern parts of the island.
The storm dropped 194 millimetres of water in the extreme southeastern province of Guantanamo on Monday, authorities said. Some reservoirs in the drought-plagued area were filled for the first time in 10 years, the island's Granma newspaper said.
Families in Florida are stocking up
on emergency gas supplies
Train services across the country were also stopped while the storm passed, and national commercial flights were suspended.
Jeb Bush, the governor of Florida, has urged residents to make preparations and not wait until the storm is upgraded to a hurricane once more.
"Take this storm very seriously. A hurricane is a hurricane," he said, urging people to have 72 hours worth of supplies.
Forecasters said the storm could go on to menace the coasts of Georgia and South Carolina.
Cruise ship companies said they were diverting several liners to avoid the storm.