Hubble Space Telescope observations of the gravitational lens PG 1115+080 in the infrared show the known z =0.310 lens galaxy and reveal the z = 1.722 quasar host galaxy. The main lens galaxy G is a nearly circular (ellipticity < 0.07) elliptical galaxy with a de Vaucouleurs profile and an effective radius of R_e = 0.59 +/- 0.06 arcsec (1.7 +/- 0.2 h^{-1} kpc for Omega = 1 and h = H_0/100 km/s/Mpc). G is part of a group of galaxies that is a required component of all successful lens models. The new quasar and lens positions (3 milliarcsecond errors) yield constraints for these models that are statistically degenerate, but several conclusions are firmly established. (1) The principal lens galaxy is an elliptical galaxy with normal structural properties, lying close to the fundamental plane for its redshift. (2) The potential of the main lens galaxy is nearly round, even when not constrained by the small ellipticity of the light of this galaxy. (3) All models involving two mass distributions place the group component near the luminosity-weighted centroid of the brightest nearby group members. (4) All models predict a time delay ratio r_{ABC} = 1.3. (5) Our lens models predict H_0 = 44 +/- 4 km/s/Mpc if the lens galaxy contains dark matter and has a flat rotation curve, and H_0 = 65 +/- 5 km/s/Mpc if it has a constant mass-to-light ratio. (6) Any dark halo of the main lens galaxy must be truncated near 1.5 arcsec (4 h^{-1} kpc) before the inferred Ho rises above 60 km/s/Mpc. (7) The quasar host galaxy is lensed into an Einstein ring connecting the four quasar images, whose shape is reproduced by the models. Improved NICMOS imaging of the ring could be used to break the degeneracy of the lens models.
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