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We report the discovery of a low-mass companion orbiting the metal-rich, main sequence F star TYC 2949-00557-1 during the MARVELS (Multi-object APO Radial Velocity Exoplanet Large-area Survey) Pilot Project. The host star has an effective temperature T_eff = 6135 +/- 40 K, log(g) = 4.4 +/- 0.1 and [Fe/H] = 0.32 +/- 0.01, indicating a mass of M = 1.25 +/- 0.09 M_\odot and R = 1.15 +/- 0.15 R_\odot. The companion has an orbital period of 5.69449 +/- 0.00023 days and straddles the hydrogen burning limit with a minimum mass of 64 M_J, and may thus be an example of the rare class of brown dwarfs orbiting at distances comparable to those of "Hot Jupiters." We present relative photometry that demonstrates the host star is photometrically stable at the few millimagnitude level on time scales of hours to years, and rules out transits for a companion of radius greater than 0.8 R_J at the 95% confidence level. Tidal analysis of the system suggests that the star and companion are likely in a double synchronous state where both rotational and orbital synchronization have been achieved. This is the first low-mass companion detected with a multi-object, dispersed, fixed-delay interferometer.