A method is provided for removing an abandoned marine platform from a seabed. The platform extends below the seabed mud line. In order to remove the platform, a frame is provided having a perimeter that surrounds an open center. The frame is lowered over the platform wherein the platform occupies the open center portion. A movable cutter cable is fitted to the frame. At least a part of the cutter cable is positioned below the mud line. The cutter is moved relative to the frame to cut the platform below the mud line. Such movement can be a back and forth movement wherein the cutter cable moves in a first direction, stops and then moves in a second direction that is generally opposite the first direction. A lifting device can employ two vessels and two frames, each frame bridging from one vessel to the other. Winches on the vessel can be used to provide movement to the cutter cable. These same winches can take up cable slack as cutting moves from one platform member (e.g. leg, jacket leg, piling, etc) to another.