  Nuclear spin-dependent parity violation (NSD-PV) effects in atoms and
molecules arise from $Z^0$ boson exchange between electrons and the nucleus,
and from the magnetic interaction between electrons and the parity-violating
nuclear anapole moment. We demonstrate measurements of NSD-PV that use an
enhancement of the effect in diatomic molecules, here using the test system
$^{138}$Ba$^{19}$F. Our sensitivity surpasses that of any previous atomic PV
measurement. We show that systematic errors can be suppressed to at least the
level of the present statistical sensitivity. We measure the matrix element,
$W$, of the NSD-PV interaction with total uncertainty $\delta W/(2\pi)<0.7$ Hz,
for each of two configurations where $W$ must have different signs. This
sensitivity would be sufficient to measure NSD-PV effects of the size
anticipated across a wide range of nuclei.
