An extremely low mass companion to the solar-type star 70 Virginis is inferred from the observed periodic Doppler reflex motion of the primary during 8 yr. The minimum mass (M2 sin i) of 70 Vir "B" is 6.6 Jovian masses (MJ), which falls in the mass range associated with "extrasolar giant planets" (0.3--15 MJ). An orbital fit to the velocities yields a period P = 116.6 days, an amplitude K = 318 m s-1, and an eccentricity e = 0.40. The residuals to the fit scatter by 8 m s-1, consistent with the errors. Thus 70 Vir B and 51 Peg B represent the first planets found outside our solar system. Alternatively, the probability that 70 Vir B is an orbiting brown dwarf of mass M > 40 MJ is ~1%, requiring an extreme (face-on) orbital inclination of i < 9 deg. With a likely mass of 6.6--9 MJ, 70 Vir B lies in the nebulously defined domain between solar system planets and brown dwarfs. Its effective temperature is computed to be ~90 degC. The formation of such giant planets in eccentric orbits is not explained by current theory.