  Using spectroscopic radial velocities with the 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 $0.075\pm0.003 \ M_{\odot}$ ($78.6\pm3.1 \ M_{Jup}$) and
$0.099^{+0.006}_{-0.004}\ R_{\odot}$ ($0.96^{+0.06}_{-0.04}\ R_{Jup}$),
respectively. Assuming the system is coeval, the evolutionary state of the
primary indicates the age is $\sim6.7$ 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 James Webb Space Telescope 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
SDSS-IV monitoring of Kepler planet candidates with the multi-object APOGEE
instrument.
