A multiple cable clamp is configured to hold a number of sheathed electrical cables in a cable run within a building structure using electrical power, with such cables in a parallel flat array. The cable clamp is configured as a pair of rail members, one being a back or distal rail member and the other being a corresponding front or proximal rail member. These rail members can favorably be formed as bars of a tough, sturdy non-conductive material such as PVC or similar plastic resin. Transverse grooves and lands in the distal member align with teeth and recesses in the proximal member to contain and secure the respective sheathed cables. The depth of the teeth can vary to accommodate different size cables.