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High-precision radial-velocity observations of the solar-type star 16 Cygni B (HR 7504, HD 186427), taken at McDonald Observatory and at Lick Observatory, have each independently discovered periodic radial-velocity variations indicating the presence of a Jovian-mass companion to this star. The orbital t to the combined McDonald and Lick data gives a period of 800.8 days, a velocity amplitude (K) of 43.9 m s~1, and an eccentricity of 0.63. This is the largest eccentricity of any planetary system discovered so far. Assuming that 16 Cygni B has a mass of 1.0 the mass function then implies a mass for the M _ , companion of 1.5/sin i Jupiter masses. While the mass of this object is well within the range expected for planets, the large orbital eccentricity cannot be explained simply by the standard model of growth of planets in a protostellar disk. It is possible that this object was formed in the normal manner with a low-eccentricity orbit and has undergone postformational orbital evolution, either through the same process that has been proposed to have formed the "" massive eccentric planets around 70 Virginis and HD 114762, or by gravitational interactions with the companion star 16 Cygni A. It is also possible that the object is an extremely low mass brown dwarf formed through fragmentation of the collapsing protostar. We explore a possible connection between stellar photospheric Li depletion, premainsequence stellar rotation, the presence of a massive protoplanetary disk, and the formation of a planetary companion.