A polish rod clamp apparatus has a body defining a lower rod aperture and a tapered body recess. A cap defines an upper rod aperture and a tapered cap recess. Tapered gripper dies have upper and lower outer surfaces sloping inward from a central portion and inner surfaces configured to bear against the polish rod. The dies are installed in the body recess such that inner surfaces bear against a polish rod and the lower outer surfaces bear against walls of the body recess. The dies are movable with respect to the body and are prevented from rotating. The cap is threaded onto the body such that the upper outer surfaces of the gripper dies bear against the walls of the tapered cap recess. Threading the cap onto the body forces the gripper dies against the outer surface of the polish rod passing through the body.