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Exact (Hartree-Fock) exchange is needed to overcome some of the limitations of local and semilocal approximations of density-functional theory. So far, however, computational cost has limited the use of exact exchange in plane-wave calculations for extended systems. We show that this difficulty can be overcome by performing a unitary transformation from Bloch to maximally localized Wannier functions in combination with an efficient technique to compute real-space Coulomb integrals. The resulting scheme scales linearly with system size. We validate the scheme with representative applications.