The rate of evolution in terms of the number of mutant substitutions in a finite population is investigated assuming a quantitative character subject to stabilizing selection, which is known to be the most prevalent type of natural selection. It is shown that, if a large number of segregating loci (or sites) are involved, the average selection coefficient per mutant under stabilizing selection may be exceedingly small. These mutants are very slightly deleterious but nearly neutral, so that mutant substitutions are mainly controlled by random drift, although the rate of evolution may be lower as compared with the situation in which all the mutations are strictly neutral. This is treated quantitatively by using the diffusion equation method in population genetics. A model of random drift under stabilizing selection is then applied to the problem of "nonrandom" or unequal usage of synonymous codons, and it is shown that such nonrandomness can readily be understood within the framework of the neutral mutation--random drift hypothesis (the neutral theory, for short) of molecular evolution.