Hurricane Barbara has maximum sustained winds of 75 mph.
The US National Hurricane Center predicts it will bring heavy rain for a prolonged period of time.
Four to 20 inches of rain is forecast, along with a storm surge of up to 5 feet above normal tide levels.
Hurricane Barbara made landfall some 80 miles east of Salina Cruz, home to Mexico’s biggest oil refinery.
A hurricane warning has been issued for the stretch of coast between Puerto Angel and Barra de Tonala.
Thousands of people in Chiapas state have been evacuated. Roads have been closed and sea traffic suspended in the area.
The storm that preceded the hurricane caused floods in many coastal areas, including the resort city of Acapulco, further north.
Hurricane Barbara is expected to lose strength and dissipate in the next 24 hours.
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