A floating side rail clamp weapon accessory mount adaptor includes a stationary side rail clamp, a floating side rail clamp, and a leaf spring system that replaces prior art biasing plates. The floating side rail clamp moves as a single unit under evenly increasing influence provided by the leaf spring system coupled to the floating side rail clamp. The floating side rail clamp moves as a single unit such that a floating side rail clamp engagement surface of the floating side rail clamp remains substantially parallel to corresponding portion of a stationary side rail clamp engagement surface of the stationary side rail clamp as the floating side rail clamp moves, as a unit, towards the stationary side rail clamp in a direction substantially perpendicular to the stationary side rail clamp engagement surface.