Clark Contracts won the work through competitive tender and the fit out is scheduled for completion by the end of February 2011.
MMS will be re-locating the 240 staff at its current Glasgow office in St Vincent Street.
Magnus Swanson, chief executive of MMS, said, "We look forward to working with Clark Contracts to deliver a new office that will offer a significantly enhanced working environment, bringing together the unique historical fabric of a stunning building within a thoroughly modern setting and a great address."
Gordon Cunningham, managing director, Clark Contracts said, "We are especially pleased to be appointed to carry out this contract for such a prestigious client and for such a historical building. Our specialist fit out team has a proven track record of successfully delivering large scale office fit out projects for both the public and private sectors."
Founded in 1871, Maclay Murray & Spens is one of Scotland's leading commercial law firms with offices in London, Edinburgh, Aberdeen and Glasgow. As the UK members of the Lex Mundi network, they have access to quality legal advice and local knowledge through 21,000 lawyers working in 100 countries worldwide.
Developed by HF Developments, G1 Glasgow cost £70m to redevelop and involved the demolition of all but the external, Grade A listed walls, with a new building constructed behind the façade, along with a further four storey glass roof extension.
As well as MMS, other occupiers at G1 are Ernst & Young, Gardiner & Theobald and leading restaurateur Jamie Oliver who opened his first Scottish 'Jamie's Italian' restaurant at G1 earlier this year.