One major goal in condensed matter is identifying the physical mechanisms that lead to arrested states of matter, especially gels and glasses. The complex nature and microscopic details of each particular system are relevant. However, from both scientific and technological viewpoints, a general, consistent and unified definition is of paramount importance. Through Monte Carlo computer simulations of states identified in experiments, we demonstrate that dynamical arrest in adhesive hard-sphere dispersions is the result of rigidity percolation with coordination number 〈n(b)〉 equal to 2.4. This corresponds to an established mechanism leading to mechanical transitions in network-forming materials [Phys. Rev. Lett. 54, 2107 (1985)]. Our findings connect the concept of critical gel formation in colloidal suspensions with short-range attractive interactions to the universal concept of rigidity percolation. Furthermore, the bond, angular, and local distributions along the gelation line are explicitly studied in order to determine the topology of the structure at the critical gel state.