  Infrared images of the Q 0957+561 gravitational lens obtained with the Hubble
Space Telescope show two large (\sim 5 arcsec) lensed images of the z_s=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 H_0. 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. Also, the cluster must be non-spherical 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.
