The 29-year-old has scored 76 goals in 278 league games during his career, including a career-best 25 during the Shots' 2007-08 promotion season.
Grant has been handed the number 28 shirt at Holker Street.
The signing follows the departure of striker Jason Walker to divisional rivals Luton Town on an initial loan deal on Tuesday.
Walker, 26, scored 64 goals in 176 games for the Bluebirds following his move from Morecambe in January 2007.
Source: BBC Sport
Source: BBC Sport