Doppler measurements of the M4 dwarf star Gliese 876 taken at both Lick and Keck Observatories reveal periodic, Keplerian velocity variations with a period of 61 days. The orbital fit implies that the companion has a mass of M=2.1 MJUP/sini, an orbital eccentricity of e=0.27+/-0.03, and a semimajor axis of a=0.21 AU. The planet is the first found around an M dwarf and was drawn from a survey of 24 such stars at Lick Observatory. It is the closest extrasolar planet yet found, providing opportunities for follow-up detection. The presence of a giant planet on a noncircular orbit, 0.2 AU from a 0.32 Msolar star, presents a challenge to planet formation theory. This planet detection around an M dwarf suggests that giant planets are numerous in the Galaxy. Based on observations obtained at Lick Observatory, which is operated by the University of California, and on observations obtained at the W. M. Keck Observatory, which is operated jointly by the University of California and the California Institute of Technology.