We report 18 years of Doppler shift measurements of a nearby star, 55 Cancri, that exhibits strong evidence for five orbiting planets. The four previously reported planets are strongly confirmed here. A fifth planet is presented, with an apparent orbital period of 260 days, placing it 0.78 AU from the star in the large empty zone between two other planets. The velocity wobble amplitude of 4.9 m s-1 implies a minimum planet mass Msin i = 45.7 M⊕. The orbital eccentricity is consistent with a circular orbit, but modest eccentricity solutions give similar χ2ν fits. All five planets reside in low-eccentricity orbits, four having eccentricities under 0.1. The outermost planet orbits 5.8 AU from the star and has a minimum mass Msin i = 3.8 MJup, making it more massive than the inner four planets combined. Its orbital distance is the largest for an exoplanet with a well-defined orbit. The innermost planet has a semimajor axis of only 0.038 AU and has a minimum mass, Msin i, of only 10.8 M⊕, making it one of the lowest mass exoplanets known. The five known planets within 6 AU define a minimum-mass protoplanetary nebula to compare with the classical minimum-mass solar nebula. Numerical N-body simulations show this system of five planets to be dynamically stable and show that the planets with periods of 14.65 and 44.3 days are not in a mean motion resonance. Millimagnitude photometry during 11 years reveals no brightness variations at any of the radial velocity periods, providing support for their interpretation as planetary. Based on observations obtained at 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.