The N=1-->0 transitions of 12CN and 13CN(X 2Σ+) at 113.5 and 108.8 GHz, respectively, have been observed in a sample of 13 Galactic molecular clouds using the Kitt Peak 12 m radio telescope. The objects studied include the Galactic center [SgrB2(OH)], sources in the solar neighborhood such as Orion A and NGC 2024, and various other clouds with and without star formation. Hyperfine structure, arising from the nitrogen nuclear spin, was resolved in the spectra of both species, enabling an accurate determination of the opacity in 12CN. From these measurements, estimates of the 12C/13C isotope ratio were obtained. These values fall in the range 12C/13C~20-70 and exhibit a noticeable gradient with distance from the Galactic center. In general, the ratios obtained from CN are very similar to those determined from millimeter observations of CO but are consistently lower than those derived from H2CO. If chemical fractionation is occurring in CN, it is comparable to that in CO. The highest 12C/13C ratios (65+/-12 and 70+/-11) were obtained toward two known photon-dominated regions (the Orion Bar and NGC 2024); CN in these two sources may be undergoing some isotope-selective photodissociation. The 12C/13C ratio of 43+/-7, found in Orion A, is similar to that determined from optical observations of CN toward a nearby source, ζ Ophiuchi, but significantly lower than the average value found in comets (90+/-10). This difference suggests that substantial enrichment of 13C has occurred during the past 4.6 Gyr in the solar neighborhood.