Massive companions in close orbits around G dwarfs are thought to undergo rapid orbital decay due to runaway tidal dissipation. We report here the discovery of WASP-128b, a brown dwarf discovered by the WASP survey transiting a G0V host on a 2.2 d orbit, where the measured stellar rotation rate places the companion in a regime where tidal interaction is dominated by dynamical tides. Under the assumption of dynamical equilibrium, we derive a value of the stellar tidal quality factor log {Q_\star ^' }} = {6.96 ± 0.19}. A combined analysis of ground-based photometry and high-resolution spectroscopy reveals a mass and radius of the host, MSUB⋆/SUB = 1.16 ± 0.04MSUB⊙/SUB, RSUB⋆/SUB = 1.16 ± 0.02RSUB⊙/SUB, and for the companion, MSUBb/SUB = 37.5 ± 0.8Mj, RSUBb/SUB = 0.94 ± 0.02Rj, placing WASP-128b in the driest parts of the brown dwarf desert, and suggesting a mild inflation for its age. We estimate a remaining lifetime for WASP-128b similar to that of some ultra-short period massive hot Jupiters, and note it may be a propitious candidate for measuring orbital decay and testing tidal theories.