Many animals navigate using a combination of visual landmarks and path integration. In mammalian brains, head direction cells integrate these two streams of information by representing an animal's heading relative to landmarks, yet maintaining their directional tuning in darkness based on self-motion cues. Here we use two-photon calcium imaging in head-fixed Drosophila melanogaster walking on a ball in a virtual reality arena to demonstrate that landmark-based orientation and angular path integration are combined in the population responses of neurons whose dendrites tile the ellipsoid body, a toroidal structure in the centre of the fly brain. The neural population encodes the fly's azimuth relative to its environment, tracking visual landmarks when available and relying on self-motion cues in darkness. When both visual and self-motion cues are absent, a representation of the animal's orientation is maintained in this network through persistent activity, a potential substrate for short-term memory. Several features of the population dynamics of these neurons and their circular anatomical arrangement are suggestive of ring attractors, network structures that have been proposed to support the function of navigational brain circuits.