  We report {\it quantitative} microwave Faraday rotation measurements
conducted with a high-mobility two-dimensional electron gas (2DEG) in a
GaAs/AlGaAs semiconductor heterostructure. In a magnetic field, the Hall effect
and the Faraday effect arise from the action of Lorentz force on electrons in
the 2DEG. As with the Hall effect, a classical Faraday effect is observed at
low magnetic field as well as a quantized Faraday effect at high magnetic
field. The high electron mobility of the 2DEG enables a giant single-pass
Faraday rotation of $\theta_F^{max} \simeq 45^\circ$ $(\simeq0.8$~rad) to be
achieved at a modest magnetic field of $B \simeq 100$~mT. In the quantum
regime, we find that the Faraday rotation $\theta_F$ is quantized in units of
$\alpha^*= 2.80(4)\alpha$, where $\alpha\simeq 1/137$ is the fine structure
constant. The enhancement in rotation quantum $\alpha^* > \alpha$ is attributed
to electromagnetic confinement within a waveguide structure.
