As well as deciding routes, SYMCA will set timetables and fares, with all revenue going to the public body.
It will also own the buses and depots and make decisions about how money is reinvested in the network.
Matt Kitchin, Stagecoach managing director, said: “We are pleased the decision on the future of bus franchising in our region has now been made.
“We look forward to continuing the conversation on what works best for customers locally, bringing our considerable experience and extensive knowledge of running services in franchised regions like London and Greater Manchester.”
During the consultation, however, TM Travel, who are based in Sheffield said they feared smaller and medium-sized bus operators could be excluded from the market.
South Yorkshire was once famous for its publicly-run bus network, with adult fares of 8p and a ‘go anywhere’ kids fare of 2p.
That ended in 1986 when the then-government deregulated the bus market outside London, allowing private companies to compete on bus routes.
Since 2017 metro mayors have been allowed to franchise bus services, with the process recently completed in Greater Manchester, where all buses have been painted a striking yellow and branded “Bee Network”.