Using spectroscopic radial velocities with the Apache Point Observatory Galaxy Evolution Experiment (APOGEE) instrument and Gaia distance estimates, we demonstrate that Kepler-503b, currently considered a validated Kepler planet, is in fact a brown-dwarf/low-mass star in a nearly circular 7.2-day orbit around a subgiant star. Using a mass estimate for the primary star derived from stellar models, we derive a companion mass and radius of 0.075 ± 0.003 M ⊙ (78.6 ± 3.1 M Jup) and {0.099}-0.004+0.006 {R}ȯ ({0.96}-0.04+0.06 R Jup), respectively. Assuming that the system is coeval, the evolutionary state of the primary indicates the age is ∼6.7 Gyr. Kepler-503b sits right at the hydrogen-burning mass limit, straddling the boundary between brown dwarfs and very low-mass stars. More precise radial velocities and secondary eclipse spectroscopy with the James Webb Space Telescope (JWST) will provide improved measurements of the physical parameters and age of this important system to better constrain and understand the physics of these objects and their spectra. This system emphasizes the value of radial velocity observations to distinguish a genuine planet from astrophysical false positives, and is the first result from the Sloan Digital Sky Survey (SDSS)-IV monitoring of Kepler planet candidates with the multi-object APOGEE instrument.