We present ALMA and MUSE observations of the Brightest Cluster Galaxy in Abell 2597, a nearby (z=0.0821) cool core cluster of galaxies. The data map the kinematics of a three billion solar mass filamentary nebula that spans the innermost 30 kpc of the galaxys core. Its warm ionized and cold molecular components are both cospatial and comoving, consistent with the hypothesis that the optical nebula traces the warm envelopes of many cold molecular clouds that drift in the velocity field of the hot X-ray atmosphere. The clouds are not in dynamical equilibrium, and instead show evidence for inflow toward the central supermassive black hole, outflow along the jets it launches, and uplift by the buoyant hot bubbles those jets inflate. The entire scenario is therefore consistent with a galaxy-spanning fountain, wherein cold gas clouds drain into the black hole accretion reservoir, powering jets and bubbles that uplift a cooling plume of low-entropy multiphase gas, which may stimulate additional cooling and accretion as part of a self-regulating feedback loop. All velocities are below the escape speed from the galaxy, and so these clouds should rain back toward the galaxy center from which they came, keeping the fountain long-lived. The data are consistent with major predictions of chaotic cold accretion, precipitation, and stimulated feedback models, and may trace processes fundamental to galaxy evolution at effectively all mass scales.