The invention relates to a cross-member to be found in the proximity of a bodywork part capable of moving towards the cross-member, said cross-member being fitted with an inertia part and a stabilization part for geometrically stabilizing the inertia part. The stabilization part is longitudinally subdivided into a first portion located beside the bodywork part, and a second portion located remote from the bodywork part. The first and second portions are designed in such a manner that in the event of the bodywork part being subjected to a force equivalent to an impact against a pedestrian, said part can move towards second portion while deforming the inertia part and the first portion, without the second portion deforming. The invention also provides a front end module including such a cross-member.