Deep Chandra observations of the Hydra A Cluster reveal a feature in the X-ray surface brightness that surrounds the 330 MHz radio lobes of the AGN at the cluster center. Surface brightness profiles of this feature and its close association with the radio lobes argue strongly that it is a shock front driven by the expanding radio lobes. The Chandra image also reveals other new structure on smaller scales that is associated with the radio source, including a large cavity and filament. The shock front extends 200–300 kpc from the AGN at the cluster center, and its strength varies along the front, with Mach numbers in the range ∼1.2–1.4. It is stronger where it is more distant from the cluster center, as expected for a shock driven by expanding radio lobes. Simple modeling gives an age for the shock front of ∼1.4 × 108 yr and a total energy driving it of ∼1061 ergs. The mean mechanical power driving the shock is comparable to quasar luminosities, well in excess of that needed to regulate the cooling core in Hydra A. This suggests that the feedback regulating cooling cores is inefficient, in that the bulk of the energy is deposited beyond the cooling core. In that case, a significant part of cluster "preheating" is a by-product of the regulation of cooling cores.