Binary pulsar systems are superb probes of stellar and binary evolution and the physics of extreme environments. In a survey with the Arecibo telescope, we have found PSR J1903+0327, a radio pulsar with a rotational period of 2.15 ms in a highly eccentric (e = 0.44) 95-day orbit around a solar mass companion. Infrared observations identify a possible main-sequence companion star. Conventional binary stellar evolution models predict neither large orbital eccentricities nor main-sequence companions around millisecond pulsars. Alternative formation scenarios involve recycling a neutron star in a globular cluster then ejecting it into the Galactic disk or membership in a hierarchical triple system. A relativistic analysis of timing observations of the pulsar finds its mass to be 1.74+/-0.04 Msun, an unusually high value.