We report the discovery of PSR J1906+0746, a young 144 ms pulsar in a highly relativistic 3.98 hr orbit with an eccentricity of 0.085 and expected gravitational wave coalescence time of ~300 Myr. The new pulsar was found during precursor survey observations with the Arecibo 1.4 GHz feed array system and retrospectively detected in the Parkes Multibeam plane pulsar survey data. From radio follow-up observations with Arecibo, Jodrell Bank, Green Bank, and Parkes, we have measured the spin-down and binary parameters of the pulsar and its basic spectral and polarization properties. We also present evidence for pulse profile evolution, which is likely due to geodetic precession, a relativistic effect caused by the misalignment of the pulsar spin and total angular momentum vectors. Our measurements show that PSR J1906+0746 is a young object with a characteristic age of 112 kyr. From the measured rate of orbital periastron advance (7.57d+/-0.03d yr-1), we infer a total system mass of 2.61+/-0.02 Msolar. While these parameters suggest that the PSR J1906+0746 binary system might be a younger version of the double pulsar system, intensive searches for radio pulses from the companion have so far been unsuccessful. It is therefore not known whether the companion is another neutron star or a massive white dwarf. Regardless of the nature of the companion, a simple calculation suggests that the Galactic birthrate of binaries similar to PSR J1906+0746 is ~60 Myr-1. This implies that PSR J1906+0746 will make a significant contribution to the computed cosmic inspiral rate of compact binary systems.