Nuclear spin-dependent parity violation arises from weak interactions between electrons and nucleons, and from nuclear anapole moments. We outline a method to measure such effects, using a Stark-interference technique to determine the mixing between opposite-parity rotational/hyperfine levels of ground-state molecules. The technique is applicable to nuclei over a wide range of atomic number, in diatomic species that are theoretically tractable for interpretation. This should provide data on anapole moments of many nuclei, and on previously unmeasured neutral weak couplings.PACS numbers: 32.80. Ys, 12.15.Mm, 21.10.Ky Up to now, atomic parity violation (PV) experiments have primarily focused on the PV effect arising from the weak charge of the nucleus Q W [1], a nuclear spinindependent quantity that parameterizes the electroweak neutral coupling between electron axial-and nucleon vector-currents (A e V n ). Here we propose a highly sensitive and widely applicable technique to measure nuclear spin-dependent (NSD) PV effects. Such effects arise primarily from two underlying causes. The first is the nuclear anapole moment, a P-odd magnetic moment induced by weak interactions within the nucleus, which couples to the spin of a penetrating electron [2]. Measurements of anapole moments can provide useful data on purely hadronic PV interactions [3,4]. So far, only one nuclear anapole moment has been measured, in 133 Cs [5]. The second source of NSD-PV is the electroweak neutral coupling between electron vector-and nucleon axialcurrents (V e A n ). This can be parameterized by two constants, C 2u,d , which describe the V e A n couplings to up and down quarks. These are suppressed in the Standard Model (SM), making C 2u,d difficult to measure and at present perhaps the most poorly characterized parameters in the SM [6]. However, because of this suppression, even moderately precise measurements of C 2u,d could be sensitive to new physics at TeV energy scales [7].Our method to measure NSD-PV exploits the properties of diatomic molecules [8,9,10] to amplify the observable signals. Rotational/hyperfine (HF) levels of opposite parity can be mixed by NSD-PV interactions, and are inherently close in energy. Accessible laboratory magnetic fields can Zeeman-shift these levels to degeneracy, dramatically enhancing the state mixing. The matrix element (m.e.) of the NSD-PV interaction can be measured with a Stark-interference technique of demonstrated sensitivity [11]. Use of ground-state molecules leads to enhanced resolution because of their long lifetimes [11,12]. (Two recent papers also proposed measuring NSD-PV in the ground state HF levels of heavy alkali atoms [13].) The technique is applicable to a wide class of molecules and hence to NSD-PV couplings to the variety of nuclei within them. We specifically consider diatomic molecules with a single valence electron in a 2 Σ electronic state. These are the molecular equivalent of alkali atoms, with a simple, regular structure of rotational/HF levels. This makes it possible to re...