A vehicle wheel and pneumatic tire assembly having a counter-weight. The assembly includes a rigid rim having essentially radially inwardly extending rim flanges, a pneumatic vehicle tire, the beads of which are mounted on the radially inner side of the rim next to the rim flanges, and respective cover rings that occupy any space disposed axially inwardly of each tire bead. In order after a counter-weight has been mounted to prevent dirt or water from penetrating between the tire bead and the cover ring, a mounting element, which serves to support the counter-weight, extends into a slot of the cover ring. The base of this slot is disposed within the cover ring.