The driver died at the scene of the crash that investigators said might have been caused by a medical event.
Initial reports indicated the bus had rolled over, but authorities later said that was not true.
The bus carrying nine elementary, middle and high school students crashed off State Road 111 near La Madera in Carson National Forest, Rio Arriba County sheriff's spokesman Jake Arnold said.
The mountainous, isolated area is popular for hunting, hiking and camping.
Two of the students who attend schools in the tiny, rural school district of Mesa Vista had broken bones.
"It is a tragedy and we are very saddened by events today," district superintendent Tracie Phillips said. "We are providing counseling for all of students, including those involved in the crash."
It was not clear if the driver, Pat Valdez, 69, might have had had a heart attack and later died from the crash, authorities said.
"We don't know if the driver died from a medical event, died from the crash or died from a combination of both," Arnold said.
The Office of the Medical Investigator will determine the cause of death, he said.
Eight students were taken to Espanola Hospital for treatment. Six were treated and released while two others were transported to University of New Mexico Hospital in Albuquerque, Espanola Hospital spokeswoman Niki Allcorn said.
One student was airlifted to UNM hospital from the scene, Allcorn said.
An 8-year-old suffered a broken back and 16-year-old had a broken jaw, authorities said.
School officials say the students were headed to Ojo Caliente Elementary, Mesa Vista Middle and Mesa Vista High schools from Vallecitos in northern Rio Arriba County.
Images from a KOB-TV helicopter showed the bus lodged in an embankment. It appears to have gone off a cliff around 15 miles from the schools in Ojo Caliente, a resort town known for its hot springs. The roads are surrounded by mesas, sagebrush and rocky terrain.
An investigation of the crash was ongoing by the Rio Arriba Sheriff's Office and New Mexico State Police. Authorities blocked off access to roads leading to the crash site.
The isolated Mesa Vista Consolidated School District is located in El Rito and is around 120 miles north of Albuquerque. It has an enrollment of close to 400 students.
Associated Press writer Jeri Clausing contributed to this story from Albuquerque, N.M.