Supermassive black holes in galaxy centres can grow by the accretion of gas, liberating energy that might regulate star formation on galaxy-wide scales. The nature of the gaseous fuel reservoirs that power black hole growth is nevertheless largely unconstrained by observations, and is instead routinely simplified as a smooth, spherical inflow of very hot gas. Recent theory and simulations instead predict that accretion can be dominated by a stochastic, clumpy distribution of very cold molecular clouds - a departure from the "hot mode" accretion model - although unambiguous observational support for this prediction remains elusive. Here we report observations that reveal a cold, clumpy accretion flow towards a supermassive black hole fuel reservoir in the nucleus of the Abell 2597 Brightest Cluster Galaxy (BCG), a nearby (redshift z=0.0821) giant elliptical galaxy surrounded by a dense halo of hot plasma. Under the right conditions, thermal instabilities can precipitate from this hot gas, producing a rain of cold clouds that fall toward the galaxy's centre, sustaining star formation amid a kiloparsec-scale molecular nebula that inhabits its core. The observations show that these cold clouds also fuel black hole accretion, revealing "shadows" cast by the molecular clouds as they move inward at about 300 kilometres per second towards the active supermassive black hole in the galaxy centre, which serves as a bright backlight. Corroborating evidence from prior observations of warmer atomic gas at extremely high spatial resolution, along with simple arguments based on geometry and probability, indicate that these clouds are within the innermost hundred parsecs of the black hole, and falling closer towards it.