A fish plate clamp for unloading rails 1. The fish plate clamp includes a support element (11, 12) with a first piece (11) and a second piece (12) each one located on a face of the rail (1). Each piece of the support element (11, 12) includes an internal mortise (13) in the area of the web (3) of the rail (1). The fish plate clamp also includes a gripping element (14) located in an internal mortise (13) of the support element (11, 12). The internal mortises (13) have an inclined surface (15) in a manner that, when the gripping element (14) moves along the length of the inclined surface (15) of the internal mortise (13) in the direction of the pull of the rail (1), the gripping element (14) latches onto the web (3) of the rail (1).