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We present a new short-period brown dwarf (BD) candidate around the star TYC 1240-00945-1. This candidate was discovered in the first year of the Multi-object APO Radial Velocity Exoplanets Large-area Survey (MARVELS), which is part of the Sloan Digital Sky Survey (SDSS) III, and we designate the BD as MARVELS-1b. MARVELS uses the technique of dispersed fixed-delay interferometery to simultaneously obtain radial velocity (RV) measurements for 60 objects per field using a single, custom-built instrument that is fiber fed from the SDSS 2.5 m telescope. From our 20 RV measurements spread over a similar to 370 day time baseline, we derive a Keplerian orbital fit with semi-amplitude K = 2.533 +/- 0.025 km s(-1), period P = 5.8953 +/- 0.0004 days, and eccentricity consistent with circular. Independent follow-up RV data confirm the orbit. Adopting a mass of 1.37 +/- 0.11 M-circle dot for the slightly evolved F9 host star, we infer that the companion has a minimum mass of 28.0 +/- 1.5 M-Jup, a semimajor axis 0.071 +/- 0.002 AU assuming an edge-on orbit, and is probably tidally synchronized. We find no evidence for coherent intrinsic variability of the host star at the period of the companion at levels greater than a few millimagnitudes. The companion has an a priori transit probability of similar to 14%. Although we find no evidence for transits, we cannot definitively rule them out for companion radii less than or similar to 1 R-Jup.