The National Hurricane Center in Miami said in its 11 p.m. EDT advisory that Ernesto, centered about 85 miles south-southwest of Veracruz and about 80 miles north of Oaxaca, was mustering maximum sustained winds of 40 mph as it slid westward at 14 mph, the U.S. forecasters said.
Ernesto's deterioration allowed the Mexican government to drop its tropical storm warning for the country's coastal region.
The hurricane center said Ernesto was expected to dissipate Friday as it moves inland over the high terrain of southern Mexico.
Still, the storm was forecast to drop another 2-5 inches of rain over the Mexican states of Veracruz, Tabasco, Puebla and Oaxaca through Friday night. Forecasters said isolated areas could be inundated with up to 15 inches, however, meaning life-threatening flash floods and mudslides would be possible.
Ernesto made landfall earlier Thursday near the city of Coatzacoals, Mexico, Thursday, sporting winds of 60 mph.
Ernesto made its first of two landfalls Tuesday night as a Category 1 hurricane on the eastern coast of Mexico's Yucatan Peninsula before advancing across southern Mexico.