Most large galaxies host supermassive black holes in their nuclei and are subject to mergers, which can produce a supermassive black hole binary (SMBHB), and hence periodic signatures due to orbital motion. We report unique periodic radio flux density variations in the blazar PKS 2131−021, which strongly suggest an SMBHB with an orbital separation of ∼0.001–0.01 pc. Our 45.1 yr radio light curve shows two epochs of strong sinusoidal variation with the same period and phase to within ≲2% and ∼10%, respectively, straddling a 20 yr period when this variation was absent. Our simulated light curves accurately reproduce the “red noise” of this object, and Lomb–Scargle, weighted wavelet Z-transform and least-squares sine-wave analyses demonstrate conclusively, at the 4.6σ significance level, that the periodicity in this object is not due to random fluctuations in flux density. The observed period translates to 2.082 ± 0.003 yr in the rest frame at the z = 1.285 redshift of PKS 2131−021. The periodic variation in PKS 2131−021 is remarkably sinusoidal. We present a model in which orbital motion, combined with the strong Doppler boosting of the approaching relativistic jet, produces a sine-wave modulation in the flux density that easily fits the observations. Given the rapidly developing field of gravitational-wave experiments with pulsar timing arrays, closer counterparts to PKS 2131−021 and searches using the techniques we have developed are strongly motivated. These results constitute a compelling demonstration that the phenomenology, not the theory, must provide the lead in this field.