The storm is expected to then move slowly north to Florida by Monday afternoon.
Charlie Crist, the governor of Florida state, declared a state of emergency, and US media reported that tourists had been advised to leave. Power is expected to be cut and businesses are likely to shut down.
"Florida is very, very practiced at these things," Crist said. "The watch word for Floridians is be calm, but be alert."
Hundreds of homes have already been brought down in the Dominican Republic and four people remain missing.