Transport Secretary Keith Brown handed responsibility for the £348m project to Network Rail yesterday, with tracks due to be in place by the end of 2014 and passenger services starting the following year on the 30-mile route between Edinburgh and Tweedbank.
Ministers had originally tried to commission a private sector contractor to build the railway – re-establishing part of the Waverley route that connected Edinburgh to Carlisle until the Beeching cuts in the 1960s – but cancelled a tendering competition last year after all but one bidder pulled out.
Network Rail will now complete the main construction project, including the creation of seven stations, at a cost of £294m.
Ministers confirmed the overall budget, including track clearance and other preparatory work, had risen to £348m – more than £50m higher than previous estimates – with the completion date slipping to summer 2015.
However, Network Rail said its involvement had helped save £60m compared to previous estimates.
The rail infrastructure company said there was a "clear target" to have the project in place by 2014, though passenger services would only commence in 2015.
Mr Brown predicted the new railway would cut car journeys on the region's roads by around 60,000 "peak trips" a year and reduce accidents on the A7 and A68 major roads.
He said: "The Borders Railway will bring inward investment for the local community plus approximately £33m of benefits for the wider Scottish economy.
"It will support 400 jobs during the construction phase and act as a catalyst for increased business development and housing opportunities within easy commuting distance of Edinburgh.
"By increasing accessibility to jobs, education and business opportunities, we can generate sustainable economic growth for the people of Midlothian, Borders and wider Scotland while offering an attractive public transport alternative to the car."
As part of the revised project, Tweedbank station will be able to accommodate tourist trains, bringing in a potentially lucrative source of income from rail enthusiasts who regularly travel on steam excursions to Scotland.
The agreement was welcomed by rail campaigners, passenger groups and business leaders.
However, Scottish Labour said there were lessons to be learned from the collapse of the procurement exercise. Richard Baker, the party's Shadow Infrastructure Secretary, said: "While any progress is to be welcomed, this is a project that has been heavily delayed, with costs spiralling £350m and a failed procurement process left behind.
"We can only hope the rest of the construction goes more smoothly, but there are significant lessons to be learned by the SNP from this project."
Network Rail, which is funding the scheme through its own borrowing powers, said the agreement followed 11 months of detailed planning and negotiations with Government agency Transport Scotland.
David Simpson, route managing director for Scotland, said: "We're delighted now to have reached an agreement on delivery of a project which will finally bring Midlothian and the central Scottish Borders back into the railway network. This agreement sets out an achiev-able delivery schedule and allows us to have trains running along the route in summer 2015.
"Network Rail has already been involved extensively with advance works and we have begun actively engaging with communities along the route to prepare communities for the impact of this major engineering challenge."