We report a detection of weak, tangential distortion of the images of cosmologically distant, faint galaxies due to gravitational lensing by foreground galaxies. A mean image polarization of (p)= 0.011 +/- 0.006 (95% confidence bounds) is obtained for 3202 pairs of source (23 < r_s_ <24) and lens (20 (20 <= r_d_ <= 23) galaxies with projected separations of 5" <= θ <= 34". Averaged over annuli of inner radIus 5" and outer radius θ_max_, the signal is string for lens-source separations of θ_max_ <~ 90" consistent with quas-isothermal galaxy halos extending to large radii (~>100 h^-1^ kpc). The observed polarization is also consistent with the signal expected on the basis of simulations incorporating measured properties of local galaxies and modest extrapolations of the observed redshift distribution of faint galaxies (to which the results are somewhat sensitive). From the simulations we obtain formal best-fit model parameters for the dark halos of the lens galaxies that consist of a characteristic circular velocity of V^*^ ~220 +/- 80 km s^-1^ and characteristic radial extent of s^*^ ~> 100 h^-1^ kpc. The predicted polarization based on the model is relatively insensitive to the characteristic radial extent of the halos, s^*^, and very small halos (s^*^ ~ 10 h^-1^ kpc) are excluded only at the 2 a level. The formal best-fit halo parameters imply typical masses for the lens galaxies within a radius of 100 h^-1^ kpc on the order of 1.0_-0.5_^+1.2^ x 10^12^ h^-1^ M_sun_ (90% confidence bounds), in agreement with recent dynamical estimates of the masses of local spiral galaxies. This is particularly encouraging as the lensing and dynamical mass estimators rely on different sets of assumptions. Contamination of the gravitational lensing signal by a population of tidally distorted satellite galaxies can be ruled out with reasonable confidence. The prospects for corroborating and improving this measurement seem good, especially using deep HST archival data.