  We examine the feasibility of a parity non-conserving (PNC) optical rotation
experiment for the $^2$P$_{3/2}\rightarrow ^2$P$_{1/2}$ transition of atomic
iodine at 1315 nm. The calculated $E1_{\rm PNC}$ to $M1$ amplitude ratio is
$R=0.80(16)\times 10^{-8}$. We show that very large PNC rotations (greater than
10 $\mu$rad) are obtained for iodine-atom column densities of $\sim 10^{22}$
cm$^{-2}$, which can be produced by increasing the effective interaction
pathlength by a factor of $\sim 10^4$ with a high-finesse optical cavity. The
simulated signals indicate that measurement of the nuclear anapole moment is
feasible, and that a 1% PNC precision measurement should resolve the
inconsistency between previous measurements in Cs and Tl.
