In overlapping genes, the same DNA sequence codes for two proteins using different reading frames. Analysis of overlapping genes can help in understanding the mode of evolution of a coding region from noncoding DNA. We identified 71 pairs of convergent genes, with overlapping 3' ends longer than 15 nucleotides, that are conserved in at least two prokaryotic genomes. Among the overlap regions, we observed a statistically significant bias towards the 123:132 phase (i.e. the second codon base in one gene facing the degenerate third position in the second gene). This phase ensures the least mutual constraint on nonconservative amino acid replacements in both overlapping coding sequences. The excess of this phase is compatible with directional (positive) selection acting on the overlapping coding regions. This could be a general evolutionary mode for genes emerging from noncoding sequences, in which the protein sequence has not been subject to selection.