We present the discovery of KELT-1b, the first transiting low-mass companion from the wide-field Kilodegree Extremely Little Telescope-North (KELT-North) survey. The V=10.7 primary is a mildly evolved, solar-metallicity, mid-F star. The companion is a low-mass brown dwarf or super-massive planet with mass of 27.23+/-0.50 MJ and radius of 1.110+0.037-0.024 RJ, on a very short period (P=1.21750007) circular orbit. KELT-1b receives a large amount of stellar insolation, with an equilibrium temperature assuming zero albedo and perfect redistribution of 2422 K. Upper limits on the secondary eclipse depth indicate that either the companion must have a non-zero albedo, or it must experience some energy redistribution. Comparison with standard evolutionary models for brown dwarfs suggests that the radius of KELT-1b is significantly inflated. Adaptive optics imaging reveals a candidate stellar companion to KELT-1, which is consistent with an M dwarf if bound. The projected spin-orbit alignment angle is consistent with zero stellar obliquity, and the vsini of the primary is consistent with tidal synchronization. Given the extreme parameters of the KELT-1 system, we expect it to provide an important testbed for theories of the emplacement and evolution of short-period companions, and theories of tidal dissipation and irradiated brown dwarf atmospheres. ; Comment: 30 pages, 19 figures. Submitted to ApJ