The Sun and >15 per cent of nearby stars are surrounded by dusty disks that must be collisionally replenished by asteroids and comets, as the dust would otherwise be depleted on timescales <10(7) years (ref. 1). Theoretical studies show that the structure of a dusty disk can be modified by the gravitational influence of planets, but the observational evidence is incomplete, at least in part because maps of the thermal infrared emission from the disks have low linear resolution (35 au in the best case). Optical images provide higher resolution, but the closest examples (AU Mic and beta Pic) are edge-on, preventing the direct measurement of the azimuthal and radial disk structure that is required for fitting theoretical models of planetary perturbations. Here we report the detection of optical light reflected from the dust grains orbiting Fomalhaut (HD 216956). The system is inclined 24 degrees away from edge-on, enabling the measurement of disk structure around its entire circumference, at a linear resolution of 0.5 au. The dust is distributed in a belt 25 au wide, with a very sharp inner edge at a radial distance of 133 au, and we measure an offset of 15 au between the belt's geometric centre and Fomalhaut. Taken together, the sharp inner edge and offset demonstrate the presence of planetary-mass objects orbiting Fomalhaut.