The bus left New Mexico 111 near La Madera and went down an embankment, the Albuquerque Journal reported. Sheriff's deputies responded to the accident call about 7:30 a.m. local time, the newspaper said.
Mesa Vista Consolidated Schools Superintendent Tracie Phillips identified the contract bus driver as Pat Valdez.
All nine students on board were taken to a hospital, most of them for observation. One student suffered a broken hip and another had a fractured jaw, the newspaper said.
Rio Arriba County Sheriff's spokesman Jake Arnold said Valdez may have suffered a heart attack or other "medical event." An autopsy was to be conducted, he said.
"Obviously this is a terrible tragedy," Phillips said. "Our thoughts and prayers go out to the family."