A bumper for a motor vehicle has a cross member disposed transversely in relation to side rails of a vehicle frame and is supported on the side rails by crash boxes. The cross member includes a central longitudinal section having opposite ends, a crash box attachment section with a substantially vertical rear wall and two legs, and an end section having in cross-section a U-shaped configuration defined by two legs and a web which interconnects the legs and forms an outward-projecting bulge. The bulge is joined with the substantially vertical rear wall of the attachment section.