Construction will start on site on the new scheme designed by Wright and Wright Architects in June and complete in early 2016.
Glenigan said Stepnell beat Beard, Feltham Construction and Kingerlee to land the job.
Stepnell will extend, refurbish and reroof the 1850s-built Grade II listed Victorian building which was originally designed by John Buckler and built as a single hall school.
In 1932 it was converted into a library for the Magdalen community supporting 200 students and 35 Fellows but student numbers have since tripled and the library needs to be revamped.
The basement of the existing Victorian building will be extended by Stepnell creating three storeys along with a new connecting one-storey extension.
The firm will also create new library areas and study rooms, flexible working spaces, a passenger lift, staircases and washing facilities.
Pete McDonald, Stepnell contracts manager, said: “We are delighted to be working with Magdalen College and respected architects Wright and Wright on this stunning project which will provide much-needed new library facilities for the academic community.
“Stepnell has significant expertise in delivering specialist schemes like this for the university sector and we’re committed to ensuring that this project creates local jobs and brings other economic benefits to the community.”