Radio Doppler data generated by the Deep Space Network (DSN) from four encounters of the Galileo spacecraft with Io, Jupiter's innermost Galilean satellite, are used to infer Io's gravitational quadrupole moments. By combining the four flybys into a single solution for the gravity field, the response of Io to the second degree tidal and rotational potentials is accurately determined. This is characterized by the value of the second degree potential Love number k2=1.2924+/-0.0027. We construct interior models that satisfy constraints imposed by the mean radius R=1821.6+/-0.5km, the mean density ρ=3527.8+/-2.9kg/m3, and the normalized axial moment of inertia C/MR2=0.37685+/-0.00035. The gravitationally derived figure of Io has principal axes (c<b<a) a=1830.0+/-0.5km, b=1819.2+/-0.5km, and c=1815.6+/-0.5km, consistent with the shape determined by imaging. Gravitational and other data strongly suggest that Io is in hydrostatic equilibrium. In this case, models of Io's interior density show that Io almost certainly has a metallic core with a radius between 550 and 900 km for an Fe-FeS core or between 350 and 650 km for an Fe core. Io is also likely to have a crust and a partially molten asthenosphere, but their thicknesses cannot be separately or uniquely determined from the gravitational data.