Somerset County Council said it had been told WebberBus would cease to operate from Friday morning "due to financial difficulties".
The authority said it would try to ensure "dozens of important bus routes" served by the company would be covered.
WebberBus managing director Tim Gardner said a statement would be issued later on Friday.
Affected services include those serving six schools, four colleges and more than a dozen public routes.
Councillor David Fothergill, who is responsible for highways, said: "WebberBus is by some distance the second largest operator in the county and organising replacement services at very short notice is a significant task."
A family-run bus company in Somerset which has been running for more than 50 years is to stop trading.