Infrared images of the Q0957+561 gravitational lens obtained with the Hubble Space Telescope show two large (~5") lensed images of the zs=1.41 quasar host galaxy. Parts of the host galaxy are doubly imaged, like the quasar, while other parts are quadruply imaged. The distortions of the host galaxy offer the best probe yet of the global structure of the lensing potential, which is essential for determining the Hubble constant from the measured time delay. The distortions are inconsistent with the predictions of previously published lens models, which invalidates those models and their implications for H0. New models show that the distortions finally break the long-standing degeneracy between the shape of the lens galaxy and the tidal shear contributed by the cluster containing the lens galaxy. The shape of the lens galaxy's mass distribution must be remarkably similar to the shape of its luminosity distribution, and most models that produce reasonable values for the Hubble constant roughly match the observed ellipticity gradient and isophote twist of the lens galaxy. In addition, the cluster must be nonspherical and produce a relatively small tidal shear. Although there are still degeneracies in the lens models that lead to a 25% uncertainty in the derived value of the Hubble constant, there are also strong prospects for new observations to further improve the constraints and reduce the uncertainties. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.