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We report the discovery of HAT-P-31b, a transiting exoplanet orbiting the V = 11.660 dwarf star GSC 2099-00908. HAT-P-31b is the first planet discovered with the Hungarian-made Automated Telescope (HAT) without any follow-up photometry, demonstrating the feasibility of a new mode of operation for the HATNet project. The 2.17 M_J, 1.1 R_J planet has a period of P_b = 5.0054 days and maintains an unusually high eccentricity of e_b = 0.2450 ± 0.0045, determined through Keck, FIbr-fed Échelle Spectrograph, and Subaru high-precision radial velocities (RVs). Detailed modeling of the RVs indicates an additional quadratic residual trend in the data detected to very high confidence. We interpret this trend as a long-period outer companion, HAT-P-31c, of minimum mass 3.4 M_J and period ⩾2.8 years. Since current RVs span less than half an orbital period, we are unable to determine the properties of HAT-P-31c to high confidence. However, dynamical simulations of two possible configurations show that orbital stability is to be expected. Further, if HAT-P-31c has non-zero eccentricity, our simulations show that the eccentricity of HAT-P-31b is actively driven by the presence of c, making HAT-P-31 a potentially intriguing dynamical laboratory.