Observations of the cosmic microwave background indicate that baryons account for 5 per cent of the Universe's total energy content(1). In the local Universe, the census of all observed baryons falls short of this estimate by a factor of two(2,3). Cosmological simulations indicate that the missing baryons have not condensed into virialized haloes, but reside throughout the filaments of the cosmic web (where matter density is larger than average) as a low-density plasma at temperatures of 10(5)-10(7) kelvin, known as the warm-hot intergalactic medium(3-6). There have been previous claims of the detection of warm-hot baryons along the line of sight to distant blazars(7-10) and of hot gas between interacting clusters(11-14). These observations were, however, unable to trace the large-scale filamentary structure, or to estimate the total amount of warm-hot baryons in a representative volume of the Universe. Here we report X-ray observations of filamentary structures of gas at 10(7) kelvin associated with the galaxy cluster Abell 2744. Previous observations of this cluster15 were unable to resolve and remove coincidental X-ray point sources. After subtracting these, we find hot gas structures that are coherent over scales of 8 megaparsecs. The filaments coincide with over-densities of galaxies and dark matter, with 5-10 per cent of their mass in baryonic gas. This gas has been heated up by the cluster's gravitational pull and is now feeding its core. Our findings strengthen evidence for a picture of the Universe in which a large fraction of the missing baryons reside in the filaments of the cosmic web.