A translationally invariant formulation of the Hartree-Fock (HF) Gamma-point approximation is presented. This formulation is achieved through introduction of the minimum image convention (MIC) at the level of primitive two-electron integrals, and implemented in a periodic version of the ONX algorithm [E. Schwegler, M. Challacombe, and M. Head-Gordon, J. Chem. Phys. 106, 9708 (1997)] for linear scaling computation of the exchange matrix. Convergence of the HF-MIC Gamma-point model to the HF k-space limit is demonstrated for fully periodic magnesium oxide, ice, and diamond. Computation of the diamond lattice constant using the HF-MIC model together with the hybrid PBE0 density functional [C. Adamo, M. Cossi, and V. Barone, THEOCHEM 493, 145 (1999)] yields a0=3.569 A with the 6-21G* basis set and a 3x3x3 supercell. Linear scaling computation of the HF-MIC exchange matrix is demonstrated for diamond and ice in the condensed phase.