We report the first reliable theoretical calculation of the quantum size correction c which yields the asymptotic ionization energy I(R) = W + (1/2 + c)/R + O(R-2) of a simple-metal cluster of radius R. Restricted-variational electronic density profiles are used to evaluate two sets of expressions for the bulk work function W and quantum size correction c : the Koopmans expressions, and the more accurate and profile-insensitive Delta SCF expressions. We find c approximate to-0.08 for stabilized (as for ordinary) jellium, and thus for real simple metals. We present parameters from which the density profiles may be reconstructed for a wide range of cluster sizes, including the planar surface. We also discuss how many excess electrons can be bound by a neutral cluster of given size. Within a continuum picture, the criterion for total-energy stability of a negatively charged cluster is less stringent than that for existence of a self-consistent solution.