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High-precision, high-cadence radial velocity monitoring over the past 8 yr at the W. M. Keck Observatory reveals evidence for a third planet orbiting the nearby (4.69 pc) dM4 star GJ 876. The residuals of three-body Newtonian fits, which include GJ 876 and Jupiter-mass companions b and c, show significant power at a periodicity of 1.9379 days. Self-consistently fitting the radial velocity data with a model that includes an additional body with this period significantly improves the quality of the fit. These four-body (three-planet) Newtonian fits find that the minimum mass of companion "d" is m sin i = 5.89 ± 0.54 M⊕ and that its orbital period is 1.93776 (±7 × 10-5) days. Assuming coplanar orbits, an inclination of the GJ 876 planetary system to the plane of the sky of ~50° gives the best fit. This inclination yields a mass for companion d of m = 7.53 ± 0.70 M⊕, making it by far the lowest mass companion yet found around a main-sequence star other than our Sun. Precise photometric observations at Fairborn Observatory confirm low-level brightness variability in GJ 876 and provide the first explicit determination of the star's 96.7 day rotation period. Even higher precision short-term photometric measurements obtained at Las Campanas imply that planet d does not transit GJ 876.