As chancellor of the University of the West of Scotland (UWS), she will represent the institution at the highest level and will have powers to award academic qualifications.
She will take over the role from Lord Smith of Kelvin, who steps down in the summer after 10 years in the post. He has held a number of prominent positions and is currently chairman of the Weir Group, Scottish and Southern Energy, Green Investment Bank and the organising committee of the Glasgow 2014 Commonwealth Games.
Dame Elish was appointed chancellor by the university's governing body yesterday and she will be formally installed at a ceremony later this year.
She is principal of St Hugh's College at the University of Oxford and is one of Scotland's most senior lawyers.
Dame Elish was the first woman and first solicitor to become the Lord Advocate, the highest law officer in the Scottish legal system.
She was awarded the DBE for services to the administration of justice in the Queen's Birthday Honours in 2011. In the same year, she was granted a Special Achievement Award by the International Association of Prosecutors.
Dame Elish said: "UWS plays a key role in the economic, social and cultural development of Scotland and I am truly honoured to accept the position of chancellor of the university.
"Universities perform a hugely important role in society and I very much look forward to being involved with the activities of UWS."
Professor Seamus McDaid CBE, principal and vice- chancellor of UWS, said: "She has demonstrated a great capacity for achieving excellence throughout her professional career and, indeed, her work embodies many of the core values of this institution. She will be a first-class ambassador for University of the West of Scotland."