We report precise Doppler velocities of HD 3651 obtained at Lick and Keck Observatories. The velocities reveal evidence of a planetary companion with an orbital period of 62.23+/-0.03 days, an eccentricity of 0.63+/-0.04, and a velocity semiamplitude of 15.9+/-1.7 m s-1. With an assumed mass of 0.79 Msolar for this K0 V star, we derive Msini=0.20MJ and a semimajor axis of 0.284 AU. The star is chromospherically inactive, with logR'HK=-5.01, and 10 years of observations at Fairborn Observatory show it to be photometrically stable to better than 0.001 mag. In particular, there is no photometric variability on the 62.23 day radial velocity period to a limit of 0.0002 mag, supporting the planetary interpretation of the radial velocity variability. Photometric transits of the planetary companion across the star are ruled out with a probability of 87%.Based on observations obtained at Lick Observatory, which is operated by the University of California, and the W. M. Keck Observatory, which is operated jointly by the University of California and the California Institute of Technology. Keck time has been granted by both NASA and the University of California.