I investigate the hypothesis that mutation rates in natural populations are determined by a balance between: (1) selection against deleterious mutations favouring lower mutation rates, and (2) selection opposing further reduction of the mutation rate, resulting from the costs incurred by more stringent proof-reading and repair (for example, a reduction in the rate of DNA replication). The influence of advantageous mutations is assumed to be negligible. In a previous paper, I analysed the dynamics of a modifier of the mutation rate in a large sexual population, where (infinitesimally rare) deleterious alleles segregate at an infinite number of unlinked loci with symmetric multiplicative fitness effects. A simple condition was obtained for a modifier allele to increase in frequency. Remarkably, this condition does not depend on the allele frequencies at the modifier locus. Here, I show that (as a consequence), given any set of possible values of the mutation rate (any set of possible modifier alleles), there always exists a single globally stable value of the mutation rate. This is an unusually strong form of "evolutionary stability" for a sexual population. Less surprisingly the optimum mutation rate in an asexual population has similar stability properties. Furthermore, in the case of an asexual population, it is not necessary to make any special assumptions about the selection acting against deleterious mutations, except that a deterministic mutation-selection equilibrium exists. I present a simple method for identifying the evolutionarily stable value of the mutation rate, given the function alpha(U) relating the value of the mutation rate to the fitness cost of maintaining this rate. I also argue that if there is a highly conserved relationship between the rate of replication per base, and the rate of mutation per base, and if this relationship has the form of a power law, then the remarkable uniformity of the per genome mutation rate in DNA based microbes can be explained.