A Chandra image of the central 100 kpc of the Abell 2597 cluster of galaxies shows bright irregular X-ray emission within the central dominant cluster galaxy (CDG) and two low surface brightness cavities located 30 kpc from the nucleus of the CDG. Unlike the cavities commonly seen in other clusters, the "ghost" cavities in Abell 2597 are not coincident with the bright central radio source. Instead, they appear to be associated with faint extended radio emission seen in a deep Very Large Array radio map. We interpret the ghost cavities as buoyantly rising relics of a radio outburst that occurred between 50 and 100 Myr ago. The demography of cavities in the few clusters studied thus far shows that galactic radio sources experience recurrent outbursts on an ∼100 Myr timescale. Over the lifetime of a cluster, ghost cavities emerging from CDGs deposit ≳1059–1061 ergs of energy into the intracluster medium. If a significant fraction of this energy is deposited as magnetic field, it would account for the high field strengths in the cooling flow regions of clusters. The similarity between the central cooling time of the keV gas and the radio cycling timescale suggests that feedback between cooling gas and the radio source may be retarding or quenching the cooling flow.