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. It has been proposed to study NSD-PV effects using an enhancement of the observable effect in diatomic molecules [D. DeMille et al., Phys. Rev. Lett. 100, 023003 (2008)]. Here, we demonstrate measurements of this type with sensitivity surpassing that of

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... any previous atomic PV measurement, using the test system ^138Ba^19F. We show that systematic errors associated with our technique can be suppressed to at least the level of the present statistical sensitivity. With ∼170 hours of data, we measure the matrix element, W, of the NSD-PV interaction with uncertainty δ W/(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.