Paula is now a tropical storm as the system hugs Cuba's northern coast and dumps heavy rain on the island.
The National Weather Service in Miami said Thursday the storm has maximum sustained winds of 70 mph (110 kph). Additional weakening is expected.
In Cuba, the storm has disrupted ferry service, turned rutted country roads into red-brown muddy quagmires and promised to lash coastal areas with dangerously high waves. Paula is expected to move inland over the island later Thursday.
A tropical storm watch is in effect for some of the Florida Keys.