Idaho state police said a preliminary investigation showed that the bus driver had a medical condition that caused the crash.
Police have identified the person killed as 33-year-old Heather Christensen of Spanish Fork, Utah. She was the school's woodwind instructor. Police said the students' injuries were not life-threatening.
The crash occurred around 7 p.m. MDT on Interstate 15, about two miles north of McCammon and 50 miles north of the Utah border, as the band headed back to American Fork High School after a competition with several other bands in Pocatello.
Miller told the Salt Lake Tribune that the teacher killed in the accident tried to avoid the crash by grabbing the steering wheel when the driver lost control and the bus veered off the road. Miller said the instructor tried to get the bus back on course, but fell out of a window as the vehicle rolled.
"She loved her students, they were her life. I know she died trying to save them," Miller told the newspaper.
Forty-nine students were on the bus that tipped over, Miller said. Altogether, a total of 222 students were on the trip, divided into four buses.
James Kimball, 16, a band member at American Fork, told the AP that he saw the bus swerve off the road and go on its side in a ditch off I-15.
"I was on the bus right behind it. We were watching a movie and I looked forward and saw the bus bouncing across the side of the road," the high school junior told the Tribune in a separate interview. "The bus flew off to the side, bumped across a couple hills and rolled over."
Distraught students sang church hymns after learning of the instructor's death, Kimball said.
Mike Huestis, the band's assistant director, told the newspaper that authorities worked to ensure that all people on the trip were accounted for and examined for injuries. Police said the bus had about 50 students, two adults and one driver on board.
Huestis traveled to the competition in his own car and was ahead of the buses. He did not see the crash, he said.
Kimball said the band, one of Utah's best known, won the Pocatello competition. The school's bands have won the Utah state title for the past 19 years. The school band also performed at President George Bush's 2005 inauguration and marched in the 2007 Macy's Thanksgiving Day parade in New York.