A new ALMA observation of the cool core brightest cluster galaxy in Abell 2597 reveals that a supermassive black hole can act much like a mechanical pump in a water fountain, driving a convective flow of molecular gas that drains into the black hole accretion reservoir, only to be pushed outward again in a jet-driven outflow that then rains back toward the galaxy center from which it came. The ALMA data reveal "shadows" cast by giant molecular clouds falling on ballistic trajectories towards the black hole in the innermost 500 parsecs of the galaxy, manifesting as deep redshifted continuum absorption features. The black hole accretion reservoir, fueled by these infalling cold clouds, powers an AGN that drives a jet-driven molecular outflow in the form of a 10 kpc-scale, billion solar mass expanding molecular bubble. The molecular shell is permeated with young stars, perhaps triggered in situ by the jet. Buoyant X-ray cavities excavated by the propagating radio source may further uplift the molecular filaments, which are observed to fall inward toward the center of the galaxy from which they came, presumably keeping the fountain long-lived. The results show that cold molecular gas can couple to black hole growth via both feedback and feeding, in alignment with "cold chaotic accretion" models for the regulation of star formation in galaxies.