Quasars have long been known to be variable sources at all wavelengths. Their optical variability is stochastic and can be due to a variety of physical mechanisms; it is also well-described statistically in terms of a damped random walk model. The recent availability of large collections of astronomical time series of flux measurements (light curves) offers new data sets for a systematic exploration of quasar variability. Here we report the detection of a strong, smooth periodic signal in the optical variability of the quasar PG 1302-102 with a mean observed period of 1,884 ± 88 days. It was identified in a search for periodic variability in a data set of light curves for 247,000 known, spectroscopically confirmed quasars with a temporal baseline of about 9 years. Although the interpretation of this phenomenon is still uncertain, the most plausible mechanisms involve a binary system of two supermassive black holes with a subparsec separation. Such systems are an expected consequence of galaxy mergers and can provide important constraints on models of galaxy formation and evolution.