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We present a new short-period brown dwarf 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 third phase of the Sloan Digital Sky Survey (SDSS-III), and we designate the brown dwarf as MARVELS-1b. MARVELS uses the technique of dispersed fixed-delay interferometery to simultaneously obtain radial velocity 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 radial velocity measurements spread over a ~370 d time baseline, we derive a Keplerian orbital fit with semi-amplitude K=2.533+/-0.025 km/s, period P=5.8953+/-0.0004 d, and eccentricity consistent with circular. Independent follow-up radial velocity data confirm the orbit. Adopting a mass of 1.37+/-0.11 M_Sun 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 instrinsic 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 ~14%. Although we find no evidence for transits, we cannot definitively rule them out for companion radii ~<1 R_Jup.