The first storm of the Atlantic hurricane season has made landfall in north-west Florida, where it is lashing the coast with heavy rain and winds.
Heavy rain fell on Miami Beach on Sunday as Alberto approached
But US weather forecasters dropped a hurricane warning for Tropical Storm Alberto as wind speeds fell from 65mph (104.5km/h) to 50mph.
Earlier Florida authorities ordered the evacuation of some 20,000 people, though reports say many stayed behind.
Alberto dropped as much as 20 inches (50cm) of rain on Cuba.
Although some Cubans were evacuated and some dilapidated buildings collapsed in the storm, there were no reports of other major damage or injuries.
Forecasters warn the storm is just the first of what is predicted to be a busy hurricane season in the US.
Alberto came ashore near Adams Beach, south-east of the state capital Tallahassee, at about 1230 (1630 GMT).
The storm scattered branches and flooded streets, but no major damage was immediately reported.
"The locals consider this a mosquito breeze," a resident from nearby Steinhatchee said.
"When it's a tropical storm, we don't even consider it," Bruce Tayco said.
Florida Governor Jeb Bush had issued mandatory evacuation orders for low-lying areas and declared a state of emergency.
Last year's record-breaking hurricane season saw the region lashed by 28 storms, 15 of which went on to become hurricanes.
The US National Oceanic and Atmospheric Administration (NOAA) has warned that activity this season will also be above normal, though it is not expected to reach last year's devastating heights.
Alberto formed off Cuba on Sunday.
The BBC's Stephen Gibbs in Cuba said TV pictures suggested Juventud island was worst affected, with most streets flooded.
Exceptionally heavy rain has also been falling in tobacco-growing areas of western Cuba.
The US hurricane season started on 1 June and lasts until 30 November.
The NOAA has predicted between 13 and 16 named storms this season, of which four are predicted to be "major storms" of category three or above.