  We investigate the weak lensing shear due to dark matter galaxy halos whose
mass distributions, as projected on the sky, are nearly elliptical. The shear
pattern due to these halos is anisotropic about the lens centers and we
quantify the level of anisotropy by comparing the mean shear experienced by
sources located closest to the major axes of the lenses $<\gamma>_{major}$, to
that experienced by sources located closest to the minor axes,
$<\gamma>_{minor}$. We also compute the degree to which the anisotropy in the
lensing signal is degraded due to a noisy determination of the position angles
of the lens galaxies and find that provided the typical error on the
orientation of the lenses is less than $15^\circ$, more than 90% of the true
lensing signal is recovered in the mean. We discuss our our results in the
context of detecting anisotropic galaxy-galaxy lensing in large, ground-based
data sets and conclude that a modest net flattening of dark matter halos should
be detectable at a statistically significant level. The forthcoming Sloan
Digital Sky Survey (SDSS) data will necessarily provide a very useful data set
for this analysis, but a detection of anisotropic galaxy-galaxy lensing is not
dependent upon the very large sky coverage of the SDSS. Rather, we argue that a
significant detection of this effect can also be obtained from an imaging
survey that is of order two magnitudes fainter than SDSS and which covers only
a relatively small area of sky, of order one square degree. [Abridged]
