  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 $\sim 0.001-0.01$ pc. Our
45.1-year radio light curve shows two epochs of strong sinusoidal variation
with the same period and phase to within $<2\%$ and $\sim 10\%$, respectively,
straddling a 20-year 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 $4.6\sigma$ significance level, that the periodicity in
this object is not due to random fluctuations in flux density. The observed
period translates to $2.082\pm 0.003$ years in the rest frame at the $z=1.285$
redshift of PKS~2131$-$021. The periodic variation in PKS~2131$-$021 is
remarkably sinusoidal. We present a model in which orbital motion, combined
with the strong Doppler boosting of the approaching relativistic jet, produces
a sine-wave modulation in the flux density which easily fits the observations.
Given the rapidly-developing field of gravitational wave experiments with
pulsar timing arrays, closer counterparts to PKS~2131$-$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.
