Nuclear spin-dependent parity violation (NSD-PV) effects in atoms and molecules arise from Z0 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 138Ba19F. 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 δW/(2π)<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.