We present 1 μm Hubble Space Telescope/near-infrared camera and multiobject spectrometer resolved imaging polarimetry of the GG Tau circumbinary ring. We find that the ring displays east-west asymmetries in surface brightness as well as several pronounced irregularities but is smoother than suggested by ground-based adaptive optics observations. The data are consistent with a 37° system inclination and a projected rotational axis at a position angle of 7° east of north, determined from millimeter imaging. The ring is strongly polarized, up to ~50%, which is indicative of Rayleigh-like scattering from submicron dust grains. Although the polarization pattern is broadly centrosymmetric and clearly results from illumination of the ring by the central stars, departures from true centrosymmetry and the irregular flux suggest that binary illumination, scattering through unresolved circumstellar disks, and shading by these disks may all be factors influencing the observed morphology. We confirm a ~025 shift between the inner edges of the near-infrared and millimeter images and find that the global morphology of the ring and the polarimetry provide strong evidence for a geometrically thick ring. A simple Monte Carlo scattering simulation is presented that reproduces these features and supports the thick-ring hypothesis. We cannot confirm filamentary streaming from the binary to the ring, also observed in the ground-based images, although it is possible that there is material inside the dynamically cleared region that might contribute to filamentary deconvolution artifacts. Finally, we find a faint fifth point source in the GG Tau field that, if it is associated with the system, is almost certainly a brown dwarf.