Submillimeter and far-infrared observations of carbon monoxide (Jaffe, Harris, and Genzel 1987; Genzel, Poglitsch, and Stacey 1988; Schmid-Burgk et al. 1989; Boreiko, Betz, and Zmuidzinas 1989) have indicated the presence of warm, dense molecular gas near regions of recent star forming activity. Estimates based on the comparison of mid-J (submm) and high-J (far-IR) 12CO lines in M17 and S106 (Harris et al. 1987a) gave a lower limit of ≈1018 cm−2 (τ(12CO 7 →6) ≈ 1) to the CO column density of quiescent (Δv ≤ 10 km/s) gas at temperatures of at least 100 K and H2 densities of 104 to 106 cm−3. The mid-J 12CO lines are likely to be optically thick in most sources. In order to obtain a better estimate of the column densities, it is thus of great interest to observe isotopic mid-J CO lines, which are likely to be optically thin.