In a sample of 114 diffuse cloud sightlines spanning a wide range of interstellar environments, we find the equivalent widths of the diffuse interstellar bands (DIBs) {lambda}6196.0 and {lambda}6613.6 to be extremely well correlated, with a correlation coefficient of 0.986. A maximum likelihood functional relationship analysis shows that the observations are consistent with a perfect correlation if the observational errors, which are dominated by continuum placement and other systematics such as interfering lines, have been underestimated by a factor of 2. The quality of this correlation far exceeds other previously studied correlations, such as that between the {lambda}5780.5 DIB and either the color excess or the atomic hydrogen column density. The unusually tight correlation between these two DIBs would seem to suggest that they might represent the first pair of DIBs known to be due to the same molecular carrier. However, further theoretical work will be required to determine whether the different linewidths and band shapes of these two DIBs can be consistent with a common carrier. If the two DIBs do not in fact share the same molecular carrier, their two carriers must be chemically very closely related.