We report on the highly extinguished afterglow of GRB 070306 and the properties of the host galaxy. An optical afterglow was not detected at the location of the burst, but in near-infrared a doubling in brightness during the first night and later power-law decay in the K band provided a clear detection of the afterglow. The host galaxy is relatively bright, R ~ 22.8. An optical low-resolution spectrum revealed a largely featureless host galaxy continuum with a single emission line. Higher resolution follow-up spectroscopy shows this emission to be resolved and consisting of two peaks separated by 7 Å, suggesting it to be [O II] at a redshift of z = 1.49594 +/- 0.00006. The infrared color H - K = 2 directly reveals significant reddening. By modeling the optical/X-ray spectral energy distribution at t = 1.38 days with an extinguished synchrotron spectrum, we derive AV = 5.5 +/- 0.6 mag. This is among the largest values ever measured for a GRB afterglow, and visual extinctions exceeding unity are rare. The importance of early near-IR observations is obvious and may soon provide a clearer view into the once elusive "dark bursts."Based on observations made at the European Southern Observatory, Paranal, Chile under programs 078.D-0416, 177.A-0591; with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias; and with the William Hershel Telescope operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.