The BBC said the bus left the road in pouring rain and fell 30 feet from a viaduct near Bath.
Remaining dates of the band's European tour have been canceled, the group's Web site said.
"It is with great regret that we have to inform you that Baroness were involved in a very serious road accident earlier today and will not be able to perform at Talking Heads tonight," the promoter of the group's planned Southampton concert Wednesday night, told the BBC. "Our thoughts are with the band at this time and we wish them and their crew a speedy recovery."
Details on who was injured, and the extent of the injuries, were not immediately disclosed to the media.
2014: The Year in Music [PHOTOS]