Glasgow Road will close for an estimated seven weeks from January 9 to allow the bridge to be demolished and a new structure built.
Work may take longer, or be postponed, if there is severe weather during this period.
During the closure, diversions will be in place via Race Road, Easton Road, Burnside Road, Mill Road and Muir Road.
Both Race Road and Muir Road will become one way.
A council spokesman said: “It is not possible to keep the road open and safely demolish and rebuild the bridge.”
Pedestrian access will be maintained throughout.