We examine 20 diffuse and translucent Galactic sight lines and extract the column densities of the ^12CO and ^13CO isotopologues from their ultraviolet A--X absorption bands detected in archival Space Telescope Imaging Spectrograph data with lambda/Deltalambda geq 46,000. Five more targets with Goddard High-Resolution Spectrograph data are added to the sample that more than doubles the number of sight lines with published Hubble Space Telescope observations of ^13CO. Most sight lines have 12-to-13 isotopic ratios that are not significantly different from the local value of 70 for ^12C/^13C, which is based on mm-wave observations of rotational lines in emission from CO and H_2CO inside dense molecular clouds, as well as on results from optical measurements of CH^+. Five of the 25 sight lines are found to be fractionated toward lower 12-to-13 values, while three sight lines in the sample are fractionated toward higher ratios, signaling the predominance of either isotopic charge exchange or selective photodissociation, respectively. There are no obvious trends of the ^12CO-to-^13CO ratio with physical conditions such as gas temperature or density, yet ^12CO/^13CO does vary in a complicated manner with the column density of either CO isotopologue, owing to varying levels of competition between isotopic charge exchange and selective photodissociation in the fractionation of CO. Finally, rotational temperatures of H_2 show that all sight lines with detected amounts of ^13CO pass through gas that is on average colder by 20 K than the gas without ^13CO. This colder gas is also sampled by CN and C_2 molecules, the latter indicating gas kinetic temperatures of only 28 K, enough to facilitate an efficient charge exchange reaction that lowers the value of ^12CO/^13CO.
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