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We report precise Doppler-shift measurements of 55 Cancri (G8 V) obtained from 1989 to 2002 at Lick Observatory. The velocities reveal evidence for an outer planetary companion to 55 Cancri orbiting at 5.5 AU. The velocities also confirm a second, inner planet at 0.11 AU. The outer planet is the first extrasolar planet found that orbits near or beyond the orbit of Jupiter. It was drawn from a sample of ~50 stars observed with sufficient duration and quality to detect a giant planet at 5 AU, implying that such planets are not rare. The properties of this Jupiter analog may be compared directly to those of the Jovian planets in our solar system. Its eccentricity is modest, e = 0.16, compared with e = 0.05 for both Jupiter and Saturn. Its mass is at least 4.0 MJUP (M sin i). The two planets do not perturb each other significantly. Moreover, a third planet of sub-Jupiter mass could easily survive between these two known planets. Indeed, a third periodicity remains in the velocity measurements with P = 44.3 days and a semiamplitude of 13 m s-1. This periodicity is caused either by a third planet at a = 0.24 AU or by inhomogeneities on the stellar surface that rotate with period 42 days. The planet interpretation is more likely, as the stellar surface is quiet both chromospherically [log(R) = -5.0] and photospherically (brightness variations less than 1 mmag). Moreover, any hypothetical surface inhomogeneity would have to persist in longitude for 14 yr. Even with all three planets, an additional planet of terrestrial mass could orbit stably at ~1 AU. The star 55 Cancri is apparently a normal, middle-aged main-sequence star with a mass of 0.95 M☉, rich in heavy elements ([Fe/H] = +0.27). This high metallicity raises the issue of the precise relationship between its age, rotation, and chromosphere.