We report the discovery of an intermediate-mass transiting brown dwarf (BD), TOI-503b, from the TESS mission. TOI-503b is the first BD discovered by TESS, and it has circular orbit around a metallic-line A-type star with a period of P = 3.6772 ± 0.0001 days. The light curve from TESS indicates that TOI-503b transits its host star in a grazing manner, which limits the precision with which we measure the BD's radius ( ${R}_{b}={1.34}_{-0.15}^{+0.26}{R}_{{\rm{J}}} ). We obtained high-resolution spectroscopic observations with the FIES, Ondřejov, PARAS, Tautenburg, and TRES spectrographs, and measured the mass of TOI-503b to be Mb = 53.7 ± 1.2 ${M}_{{\rm{J}}} . The host star has a mass of M⋆ = 1.80 ± 0.06 M☉, a radius of R⋆ = 1.70 ± 0.05R☉, an effective temperature of Teff = 7650 ± 160 K, and a relatively high metallicity of 0.61 ± 0.07 dex. We used stellar isochrones to derive the age of the system to be ∼180 Myr, which places its age between that of RIK 72b (a ∼10 Myr old BD in the Upper Scorpius stellar association) and AD 3116b (a ∼600 Myr old BD in the Praesepe cluster). Given the difficulty in measuring the tidal interactions between BDs and their host stars, we cannot precisely say whether this BD formed in situ or has had its orbit circularized by its host star over the relatively short age of the system. Instead, we offer an examination of plausible values for the tidal quality factor for the star and BD. TOI-503b joins a growing number of known short-period, intermediate-mass BDs orbiting main-sequence stars, and is the second such BD known to transit an A star, after HATS-70b. With the growth in the population in this regime, the driest region in the BD desert ( $35\mbox{--}55{M}_{{\rm{J}}}\sin i ) is reforesting.