We develop a stable scheme for full spin projection of an unrestricted Hartree-Fock wave function using ideas originally discussed by Lowdin and Harriman. We examine three test cases within the intermediate neglect of differential overlap/spectroscopic method (INDO/S); a model of ferredoxin, Fe-S enzyme models for nitratases, and Ni-6 clusters. The projected INDO/S UHF method has a remarkable ability to predict spin multiplicities correctly and with far greater ease than corresponding restricted open-shell Hartree-Fock calculations followed by configuration interaction (ROHF/CI).