A damping device includes a cylinder which is under pressure and which has a connection opening for a mount-on part which is fastened to an outer surface of the cylinder by means of an annular tensioning band element. The tensioning band element encircles the cylinder and the mount-on part on a circumferential region and, by its pretensioning, exerts a holding force on the mount-on part. The tensioning band element is underdimensioned in relation to the cross section of the cylinder and mount-on part and the distance between the two structural component parts and is expanded in a final mounting position in relation to a pre-mounting state in order to apply the pretensioning.