The storm lost much of its power as it swirled over mountainous areas of eastern Cuba, leading weather experts to revise earlier forecasts that it would regain hurricane status before reached Florida.
Projections by the National Hurricane Center (NHC) suggest the tropical storm would slam ashore near Miami by early Wednesday.
The NHC says that after a couple of hours in Florida Ernesto could head back out to the Atlantic and come ashore again in South Carolina.
By the time it hits South Carolina, the storm could regain the strength it reached on Sunday when it became this year's first hurricane.
Emergency declared
Florida Governor Jeb Bush insisted that in any case, residents should brace for possible trouble and stressed forecasts could still change.
"Take this storm very seriously," said Governor Bush. He has declared a state of emergency and put National Guard troops on standby for possible rescue operations.
Schools in Miami have been closed and authorities have ordered visitors on the Florida Keys chain of islands to head to the US mainland.
Officials told people elsewhere to get ready for possible evacuations and to store up on essential items.
NASA has postponed the planned blast-off of Atlantis and is returning the space shuttle to its hangar at the Kennedy Space Center on south-eastern Florida's Atlantic coast, close to the storm's projected track.
The tropical storm dumped rain on eastern Cuba on Monday, raising fears of flash floods and mudslides, but was far weaker than originally anticipated, with maximum sustained winds of 65 kilometres an hour.
At 2100 GMT it neared Cuba's eastern coast after crossing the width of the island.
Ernesto is expected to regain power as it churns over the warm waters toward storm-weary Florida, which was hit by eight hurricanes in the past two years.