* Storm expected to become a hurricane by Tuesday
* Forecast takes Ernesto north of Honduras, into Mexico
* Tropical Storm Florence fizzles over open Atlantic (Changes dateline, adds details from Honduras)
TEGUCIGALPA, Aug 6 (Reuters) - Tropical Storm Ernesto picked up speed in the western Caribbean on Monday as it moved toward Mexico's Yucatan Peninsula, dousing Honduras and Belize on its way.
U.S. forecasters said Ernesto had top sustained winds of 65 miles per hour (100 km per hour) on Monday afternoon and could cross the 74 mph (119 kph) threshold to become a hurricane by Tuesday morning.
Hurricane warnings were issued for part of the Yucatan's east coast and the entire coast of Belize. Tropical storm watches and warnings were in effect for other parts of the Yucatan, Honduras and the Bay Islands.
Ernesto was centered about 295 miles (475 km) east of Honduras' Roatan Island and was moving west-northwest at 12 miles per hour, according to the National Hurricane Center's 8:00 p.m. EST advisory.
Heavy rains fell over coastal regions in Honduras, including Colon, Islas de la Bahia, Gracias a Dios and Atlantida.
As a precaution, Honduran authorities ordered fishing boats to return to the port of Cortes. The NHC said rainfall of 3 to 5 inches was likely along the northern coast of Honduras and north-eastern Nicaragua, with up to 8 inches possible.
The government of Belize declared a state of alert and urged residents on vulnerable islands to move inland and those in flood-prone areas to seek shelter. Residents reported only light rain so far.
"On the forecast track, the center will be passing north of the coast of Honduras tonight and Tuesday and be near the east coast of the Yucatan Peninsula early Wednesday," forecasters at the U.S. National Hurricane Center in Miami said on Monday.
Ernesto's outer bands brought winds and rain to Jamaica as it passed south of the island on Sunday, but the storm failed to dampen street celebrations there for sprinter Usain Bolt's victory in the 100 meters track final at the Olympics Games.
Heavy rains also lashed Hispaniola and Puerto Rico on Sunday.
The forecasters expect Ernesto to move into the southern part of the Gulf of Mexico by Thursday, but it was too early to know if it could disrupt oil and gas operations in the gulf.
To the east, Florence, the sixth named storm of the Atlantic-Caribbean hurricane season, dissipated into a remnant low pressure area over the open Atlantic on Monday. It was about midway between the coast of Africa and the Leeward Islands and never threatened land.
August and September are usually the most active months of the Atlantic-Caribbean hurricane season, which runs from June 1 to Nov. 30. (Reporting by Gustavo Palencia in Tegucigalpa, Mike McDonald in Guatemala City, David Alire Garcia in Mexico City and Jane Sutton, David Adams and Michael Connor in Miami; Editing by Anthony Boadle)