The water vapor (H2O) maser source in the nucleus of NGC 3079 has been observed with a four-station intercontinental VLBI array, which had a resolution of 0.3 mas. The angular separation among the individual maser features is ∼0.3 mas (7 × 1016 cm), but individual features are unresolved and smaller than 0.06 mas (1.5 × 1016 cm). We also made VLA observations of NGC 3079, which show that the maser source is coincident in angle with the compact continuum radio source in the nucleus of the galaxy to better than 5 mas. The water vapor maser source in NGC 3079 is comparable in size, but apparently 500 times more luminous than the brightest H2O maser source in the Galaxy, W49N, based on the assumption of isotropic emission. The small size and apparent high luminosity of this extragalactic source suggests special circumstances for the maser. A pumping model with many OB stars confined to such a very small volume is implausible. We propose a model wherein the maser is an unsaturated amplifier of the nuclear continuum source. The maser has a low gain of ∼80, and the maser emission is very highly beamed. The true luminosity of a single masing cloud with its radiation beamed toward us can be ∼10-7 L⊙. The pumping requirements for a single cloud, ∼1048 pump photons s-1, are about the same as for a strong maser in our Galaxy. Based on probability arguments for detecting extragalactic masers, the total luminosity of all masers in NGC 3079 can be on the order of 10-50 L⊙.