The deal is the second largest ever completed by Stagecoach, already the UK's biggest bus and rail operator, and brings it one step closer to its ambition of becoming an international transport group.
Brian Souter, the executive chairman of Stagecoach who holds shares valued at over pounds 400m, said the deal gave the group opportunities in a consolidating US bus sector. It would also give Stagecoach a presence in a market where rail privatisation was "moving up the agenda", he said.
In spite of being the largest US bus operator, Coach USA has only 2 per cent of a market in which more than 5,000 bus companies operate.
Growing by acquisition, the Houston group has bought more than 70 smaller companies since it listed in New York in 1995.
Shares in Coach USA have plummeted since last year, falling from a peak of $45 to less than $18. US investors were concerned that the group lacked the financial muscle to be a consolidator in the US market.
Stagecoach will pay $42 a share for the US company, a 41 per cent premium to its value before 7 June when takeover speculation first took hold.
Larry King, chairman and chief executive of Coach USA, will keep his management role. He and his fellow board members have pledged a total of 8 per cent of the group's shares to the Stagecoach offer.
Mr King said he had spoken to other bidders for the group, but he played down the possibility of a counter-bid. Coach USA has promised to pay a $35m fee should another bidder emerge.
Mike Kinski, group chief executive of Stagecoach, said the deal allowed Stagecoach to cash in on the $40bn bus and train market in the US.
The US coach market has been boosted by growing numbers of US pensioners, a big source of custom for coach tour operators.
Municipal governments are also outsourcing light rail and bus operations to private operators. School buses, still 90 per cent owned by the state, are slowly being privatised.
Until now, only one other UK transport company, National Express, has made acquisitions in the US.
"What Coach USA offers at the moment is a tremendous opportunity to enter the market at a very exciting stage of its consolidation," Mr Kinski said.
The City yesterday welcomed the deal, marking Stagecoach shares up 3p to 226.5p.