Binary and multiple star systems are a frequent outcome of the star formation process and as a result almost half of all stars with masses similar to that of the Sun have at least one companion star. Theoretical studies indicate that there are two main pathways that can operate concurrently to form binary/multiple star systems: large-scale fragmentation of turbulent gas cores and filaments or smaller-scale fragmentation of a massive protostellar disk due to gravitational instability. Observational evidence for turbulent fragmentation on scales of more than 1,000 astronomical units has recently emerged. Previous evidence for disk fragmentation was limited to inferences based on the separations of more-evolved pre-main sequence and protostellar multiple systems. The triple protostar system L1448 IRS3B is an ideal system with which to search for evidence of disk fragmentation as it is in an early phase of the star formation process, it is likely to be less than 150,000 years old and all of the protostars in the system are separated by less than 200 astronomical units. Here we report observations of dust and molecular gas emission that reveal a disk with a spiral structure surrounding the three protostars. Two protostars near the centre of the disk are separated by 61 astronomical units and a tertiary protostar is coincident with a spiral arm in the outer disk at a separation of 183 astronomical units. The inferred mass of the central pair of protostellar objects is approximately one solar mass, while the disk surrounding the three protostars has a total mass of around 0.30 solar masses. The tertiary protostar itself has a minimum mass of about 0.085 solar masses. We demonstrate that the disk around L1448 IRS3B appears susceptible to disk fragmentation at radii between 150 and 320 astronomical units, overlapping with the location of the tertiary protostar. This is consistent with models for a protostellar disk that has recently undergone gravitational instability, spawning one or two companion stars.