Ewald sum simulations of the solvation thermodynamics of charged solutes employ an unphysical uniform neutralizing background charge to ensure convergence of the periodic lattice sum. Continuum corrections to the solvation free energy, entropy, and volume of single ions and ion pairs for the effect of the neutralizing background and periodic boundary conditions at finite cell size are derived in order to allow efficient calculations of the ionic properties at infinite dilution. The derivation presented in this paper shows the physical origin of the effects and can be easily extended to multiple charge sites. Corrections are small for high dielectric constant solvents but become increasingly important as the ion size is increased, the dielectric constant is decreased, or the unit cell size is decreased. An alternative way of calculating the thermodynamic properties from Ewald sum simulations is proposed for which the corrections are small for low dielectric constant solvents. Tests for low and high dielectric constant water show that, after the appropriate continuum corrections are applied, the free energy of charging an ion using Ewald sum simulations agrees with the results for potential truncation simulations and spherical boundary simulations (when corrected for truncation effects). The corrected free energy of hydration is not sensitive to the system size even for low dielectric constants. The continuum model correctly predicts that Ewald sum simulations yield a solvent polarization at large distance from an ion that is smaller than the polarization of a truly infinite system.