Forecasters in the United States said earlier on Sunday that Ernesto could become a Category Three hurricane with 185kph winds.
The US National Hurricane Centre (NHC) said: "Latest recon data indicate Ernesto is likely undergoing rapid intensification."
But forecasters downgraded Ernesto when its sustained winds slowed from 120kph to less than 97kph as mountainous terrain in southern Haiti disrupted the storm.
Haiti is a largely deforested island where previous tropical storms and hurricanes have killed thousands of people by triggering flash floods and mudslides.
Forecasters said that Haiti, the poorest country in the Americas, could see up to 50cm of rain.
Jacques Edouard Alexis, Haiti's prime minister, said that emergency supplies were being sent to affected areas, UN peacekeepers were being mobilised and about 25 families were being moved from a flood-prone slum in the capital.
He said: "The cyclone is moving slowly and we are very concerned that the consequences could be serious.
"That's why the government is mobilised at the highest level and measures have been taken to limit possible damage."
Ernesto is expected to be near the southeastern coast of Cuba on Monday morning.
Cuba began evacuating 200,000 people from its eastern provinces and called its fishing fleet to harbour as Ernesto swept through the Caribbean Sea just south of Haiti.
The NHC said Ernesto could weaken as it moves across Cuba on Monday but was expected to regain strength and become a Category Two hurricane with sustained winds of about 145kph in the Gulf.
Max Mayfield, the NHC director, said: "I expect it to come off the north coast of Cuba as a tropical storm and then ... over the Florida Straits we think that it has a very good chance of intensifying again."
The NHC said that
the storm would probably come ashore on Florida's west coast near Tampa on Thursday.
Jeb Bush, Florida's governor, ordered a state of emergency, saying the southeastern state "may be threatened by a major disaster".
Authorities ordered all visitors and non-residents in the Florida Keys to head to safety.