We investigate nuclear magnetic resonance (NMR) in near zero field, where the Zeeman interaction can be treated as a perturbation to the electron mediated scalar interaction (J coupling). This is in stark contrast to the high-field case, where heteronuclear J couplings are normally treated as a small perturbation. We show that the presence of very small magnetic fields results in splitting of the zero-field NMR lines, imparting considerable additional information to the pure zero-field spectra. Experimental results are in good agreement with first-order perturbation theory and with full numerical simulation when perturbation theory breaks down. We present simple rules for understanding the splitting patterns in near-zero-field NMR, which can be applied to molecules with nontrivial spectra.