Structural and thermodynamic properties of ionic fluids are related to those of a simpler "mimic" system with short ranged intermolecular interactions in a spatially varying effective field by use of local molecular field (LMF) theory, already successfully applied to nonuniform simple fluids. By consistently using the LMF approximation to describe only the slowly varying part of the Coulomb interaction, which we view as arising from a rigid Gaussian charge distribution with an appropriately chosen width a, exceptionally accurate results can be found. In this paper we study a uniform system of charged hard spheres in a uniform neutralizing background, where these ideas can be presented in their simplest form. At low densities the LMF theory reduces to a generalized version of the Poisson-Boltzmann approximation, but the predicted structure factor satisfies the exact Stillinger-Lovett moment conditions, and with optimal choice of or the lowest order approximation remains accurate for much stronger couplings. At high density and strong couplings the pair correlation function in the uniform mimic system with short ranged interactions is very similar to that of the full ionic system. A simple analytic formula can then describe the difference in internal energy between the ionic system and the associated mimic system.