The M62 works will take place between junctions 25 and 30 (Brighouse to Oulton), starting in the last quarter of 2011 and with completion due in 2013-14.
Work will include adapting sections of the hard shoulder with resurfacing and drainage improvements for use as either permanent or peak period additional lanes.
The contract is part of the Highways Agency's four-year National Major Projects Framework to deliver managed motorways across England. bmJV secured a place on the £2bn framework in February 2010.
Morgan Sindall Group chief executive Paul Smith said: "This award underlines our strength in the highways sector, where we have an excellent track record and are already delivering a range of projects to modernise the motorway network in the UK.
"It also highlights the value of securing positions on major frameworks, which help to provide visibility and security of earnings."