Previously, it has been recognized that radio halos in galaxy clusters are preferentially associated with merging systems, as indicated by substructure in the X-ray images and temperature maps. Since, however, many clusters without radio halos also possess substructure, the role of mergers in the formation of radio halos has remained unclear. By using power ratios to relate gravitational potential fluctuations to substructure in X-ray images, we provide the first quantitative comparison of the dynamical states of clusters possessing radio halos. A correlation between the 1.4 GHz power (P1.4) of the radio halo (or relic) and the magnitude of the dipole power ratio (P1/P0) is discovered such that approximately P1.4∝P1/P0; i.e., the strongest radio halos appear only in those clusters currently experiencing the largest departures from a virialized state. From the additional consideration of a small number of highly disturbed clusters without radio halos detected at 1.4 GHz and recalling that radio halos are more common in clusters with high X-ray luminosity (Giovannini, Tordi, & Feretti), we argue that radio halos form preferentially in massive (LX≳0.5x1045 ergs.s–1) clusters experiencing violent mergers (P1/P0≳0.5x10–4) that have seriously disrupted the cluster core. The association of radio halos with massive, large-P1/P0, core-disrupted clusters can account for both the vital role of mergers in accelerating the relativistic particles responsible for the radio emission as well as the rare occurrence of radio halos in cluster samples.