  Faraday rotation has become a powerful tool in a large variety of physics
applications. Most prominently, Faraday rotation can be used in precision
magnetometry. Here we report the first measurements of gyromagnetic Faraday
rotation on a dense, hyperpolarized $^3$He gas target. Theoretical calculations
predict the rotations of linearly polarized light due to the magnetization of
spin-1/2 particles are on the scale of 10$^{-7}$ radians. To maximize the
signal, a $^3$He target designed to use with a multipass cavity is combined
with a sensitive apparatus for polarimetry that can detect optical rotations on
the order of 10$^{-8}$ radians. Although the expected results are well above
the sensitivity for the given experimental conditions, no nuclear-spin induced
rotation was observed.
