A survey of the ^12^C/^13^C ratio toward circumstellar envelopes has been conducted at millimeter wavelengths using the facilities of the Arizona Radio Observatory (ARO). The ratios were obtained for a sample of 18 local C- and O-rich asymptotic giant branch and supergiant stars from observations of the ^12^C and ^13^C isotopologues of CO and CN, respectively. The J=1->0 transitions of both molecules were observed at {lambda}=3mm using the ARO 12m telescope, while the J=2->1 lines of the two species were measured using the ARO Sub-Millimeter Telescope (SMT) at {lambda}=1mm. The ^12^C/^13^C ratios were determined from the CO data by modeling both transitions simultaneously with a circumstellar radiative transfer code, which can account for the high opacities present in the emission from this species. In the case of CN, the hyperfine structure was used to evaluate opacity effects. Ratios obtained independently from CO and CN are in good agreement.