The fixation of a new allele can be driven by Darwinian positive selection or can be due to random genetic drift. Identifying instances of positive selection is a difficult task, because its impact is routinely obscured by the action of negative selection. The nature of the genetic code dictates that positive selection in favor of an amino acid replacement should often cause a burst of two or three nucleotide substitutions at a single codon site, because a large fraction of amino acid replacements cannot be achieved after just one nucleotide substitution. Here, we study pairs of successive nonsynonymous substitutions at one codon in the course of evolution of HIV-1 genes within HIV-1 populations inhabiting infected individuals. Such pairs are more numerous and more clumped than expected if different substitutions were independent and than what is observed for pairs of successive synonymous substitutions. Bursts of nonsynonymous substitutions in HIV-1 evolution cannot be explained by mutational biases and must, therefore, be due to positive selection. Both reversals, exact or imprecise, of fixed deleterious mutations and acquisitions of amino acids with new properties are responsible for the bursts. Temporal clumping is strongest at codon sites with a low overall rate of nonsynonymous evolution, implying that a substantial fraction of replacements of conservative amino acids are driven by positive selection. We identified many conservative sites of HIV-1 proteins that occasionally experience positive selection.