An arrangement for fastening a bumper to a vehicle includes a deformation organ having one end for attachment to the bumper, and another end for sliding into a side member. The insertion section includes an inner wall which is stretchable when subjected to a tensile force. For elongating the inner wall, a material strip is received in a receiving space and equipped with a threaded breakthrough, which is aligned with a breakthrough of the adjacent transverse wall for receiving a tensile screw. The tensile screw extends through a breakthrough in a side member and exerts a force which elongates the inner wall. This initially provides a loose sliding-in of the insertion section in the side member and, following stretching of the inner wall, a tight seat of the insertion section in the side member.