The J = 1 to J = 0 transition of carbon monoxide is used to study the abundance ratios of carbon and oxygen isotopes in dense molecular clouds. Very high-quality spectra were obtained of (C-12)(O-16), (C-13)(O-16), and (C-12)(O-18) in 14 regions distributed throughout the galactic plane. Effects of line saturation are carefully examined, and it is found that the (C-13)O species often produces a slightly saturated line. When saturation is properly taken into account, the data from all the regions cluster about a single value of approximately 14 for the (C-13)(O-16)/(C-12)(O-18) abundance ratio. This value is significantly different from the corresponding terrestrial ratio of 5.6, and it is suggested that the difference results from chemical evolution having occurred in the Galaxy since the birth of the solar system. Results are presented for additional isotope studies which indicate that the value of (C-12)/(C-13) is about 40, as compared with the terrestrial value of 89.