Borough will serve contemporary European cuisine in a “relaxed” 38-cover dining room.
The menu will offer large and small plates designed to be enjoyed in different combinations, alongside a broad selection of wines – many served by the glass.
Murray, who cooked at Norn from 2016, will head up the kitchen while Archie Kyle, formerly of Fishers in Leith, will be restaurant manager.
Murray says Borough will fit "comfortably" in to the neighbourhood, "combining a friendly, convivial atmosphere with contemporary cuisine inspired by seasonal sourcing and partnerships with local suppliers".
Norn, which closed earlier this year, attracted rave reviews from national critics including Marina O’Loughlin and the late AA Gill. Former head chef Scott Smith has since opened Edinburgh restaurant Fhior with his wife Laura.
Borough's launch comes amid a flurry of restaurant openings in the Edinburgh area. The last few months has seen the launch of The Little Chartroom (from two Castle Terrace alumi), a Roux family restaurant at The Balmoral Hotel, and 83 Hanover Street (from former Gleneagles restaurant manager Juan Jose Castillo Castro).
MasterChef: The Professionals finalist Chris Niven is heading up an all-day opening at The Scotsman Hotel, which launches this month.