The Projector Augmented Wave (PAW) formalism developed by Bl\"{o}chl [Phys. Rev. B {\bf{50}}, 17953 (1994)] has been demonstrated to be an accurate and efficient pseudo-potential-like scheme for electronic structure calculations within density functional theory. We have extended this formalism to treat the integral-differential equations of Hartree-Fock (HF) theory, demonstrating that the PAW-HF method is able calculate valence energies with the same accuracy as the frozencore orbital approximation. We show that electrons in core states provide significant contributions to the valence exchange energy and we develop and compare two schemes for incorporating their effects into the PAW-HF formalism.