Just before landfall, the National Hurricane Center lowered its warning for Alberto from hurricane to tropical storm. Alberto then came ashore shortly after noon near Adams Beach, southeast of Tallahassee.
Before daybreak emergency crews were directing a few lonely cars around flooded streets, dreading a possible 10-foot storm surge in a foot of rain, reports CBS News correspondent Jim Acosta.
More than 20,000 people along Florida's Gulf Coast were ordered to evacuate, but officials worried some residents in low-lying areas prone to flooding still would not take the storm seriously.
"A lot of people aren't going to leave," said Jackie Gorman, Cedar Key's community development director. "We're hoping this is going to be a small one, but who knows."
But a lot of people did heed the warnings.
This storm was a sort of a trial run for coastal residents now that hurricane season is here, reports CBS News correspondent Cami McCormick in Chiefland. Many followed the advice and headed inland, and there was no shortage of gasoline or other hurricane supplies.
At 11 a.m. EDT, Alberto was centered about 50 miles southeast of Tallahassee, just offshore of Keaton Beach, Fla., was moving northeast at about 9 mph, the National Hurricane Center said. Its top sustained winds were at 50 mph; the minimum for a hurricane is 74 mph.
The top wind gust hit 60 mph early Tuesday in Tampa, and about 4 to 6 inches of much-needed rain had fallen in areas that had been dry, said Charles Paxton, a meteorologist at the National Weather Service in Ruskin. There were reports of limited power outages, minor damage and fallen trees in the Tampa Bay area.
In St. Marks, Fla., Dave Cohen of CBS radio affiliate WWL-AM reports heavy rain was blanketing the region with squall lines of sideways rain. But period breaks in the deluge has allowed the water to run off, and Cohen saw little standing water.
Gov. Jeb Bush signed a declaration of emergency Monday allowing him to call up the National Guard and put laws against price gouging in place.
"We don't want to overdo it. It's not a Katrina or a Wilma, but storm surge and flooding could still cause loss of life," said Max Mayfield, the hurricane center's director.
The tropical depression that produced Alberto formed Saturday, nine days after the June 1 start of the hurricane season. The storm's winds accelerated with startling speed Monday from 50 mph to 70 mph in just three hours.
Evacuation orders were posted for people in mobile homes or low-lying areas in at least five coastal counties stretching more than 100 miles. Those ordered evacuated included about 21,000 residents of Citrus, Levy and Taylor counties.
Forecasters said it could bring 4 to 10 inches of rain to central Florida and southeastern Georgia. Rain started falling Monday and at least two tornadoes had formed, though there were no reports of any injuries or damage.
In Florida, homeowners gassed up their vehicles and stocked up on chain saws, plywood and other emergency supplies. Workers at a marina in St. Petersburg said they planned to work through the night securing more than 600 boats.
"We try and prepare as best we can but, you know, we're just not ready for it," Cedar Key resident Sherry Colson told CBS News. "I don't want it to start so soon."
Alberto also prevented the crew of space shuttle Discovery from flying Monday to the Kennedy Space Center from Houston for several days of dress rehearsals for their expected launch in July.
Wal-Mart Stores Inc. ramped up its emergency operations center Monday for the kind of disaster relief effort that won it praise for responding faster than the government last year after Hurricane Katrina.
On Monday, Alberto drenched western Cuba after a weekend of heavy rains prompted evacuations, caused some dilapidated buildings to collapse and flooded low-lying areas in Havana. There were no reports of other major damage or injuries.
Scientists say the 2006 season could produce as many as 16 named storms, six of them major hurricanes. Last year's hurricane season was the most destructive on record and the busiest in 154 years of storm tracking, with a record 28 named storms and a record 15 hurricanes.