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 (M(2) sin i) of 70 Vir ''B'' is 6.6 Jovian masses (M(J)), which falls in the mass range associated with ''extrasolar giant planets'' (0.3-15 M(J)). 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 M(J) is similar to 1%, requiring an extreme (face-on) orbital inclination of i < 9 degrees. With a likely mass of 6.6-9 M(J), 70 Vir B lies in the nebulously defined domain between solar system planets and brown dwarfs. Its effective temperature is computed to be similar to 90 degrees C. The formation of such giant planets in eccentric orbits is not explained by current theory.