  Electron interactions with the nuclear-spin-dependent (NSD) parity
non-conserving (PNC) anapole moment are strongly enhanced within heteronuclear
diatomic molecules. A novel, low-energy optical rotation experiment is being
proposed with the aim of observing NSD PNC interactions in HgH. Based on the
relativistic coupled cluster method we present a complete calculation of the
circular polarization parameter $P = 2\,\textrm{Im}(E1_{PNC})/M1 \approx 3
\times 10^{-6}\; \kappa$ for the $^2\Sigma_{1/2} \to ^2\Pi_{1/2}$ optical
transition of HgH, where $\kappa$ is a dimensionless constant determined by the
nuclear anapole moment. This provides an improvement in sensitivity to NSD PNC
by 2 -- 3 orders of magnitude over the leading atomic Xe, Hg, Tl, Pb and Bi
optical rotation experiments, and shows that the proposed measurement will be
sensitive enough to extract the $^{199}$Hg anapole moment and shed light on the
underlying theory of hadronic parity violation.
