We have observed the (CO)-C-12 and (CO)-C-13 J = 9-8 transitions at a spectral resolution of 1.5 km s-1 in four Galactic molecular clouds: W3, DR 21, W51, and M17. The line shapes are generally similar to those of low-J CO lines for each species, with the exception that line reversals produced by less excited foreground gas are not evident. The ratio of (CO)-C-12 and (CO)-C-13 intensities shows that for all sources the J = 9-8 (CO)-C-12 line is optically thick with tau-approximately 5-20. This high optical depth requires a significant amount of dense (n(H-2) > 10(5) cm-3) gas. The peak brightness temperature of the (CO)-C-12 J = 9-8 line is similar to or less than the dust temperature for all sources, and most of the detected J = 9-8 radiation appears to originate in the cool (T approximately 50 K) molecular gas which comprises the bulk of the clouds. Non-LTE modeling suggests that the gas giving rise to the J = 9-8 radiation is clumped on a scale smaller than our 80" beam, and that part of the radiation arises from a warmer (T greater-than-or-equal-to 100 K) component which is very beam diluted. We do not detect the large amount of hot (T approximately 200-500 K), spatially extended gas that has been inferred from previous observations of the (CO)-C-12 J = 7-6 line. In addition to a narrow component, the (CO)-C-12 J = 9-8 line shows evidence for a broad emission component (DELTA-V approximately 18 km s-1) in W51 and possibly DR 21. The temperatures and densities of the regions emitting the broader emission appear similar to those producing the dominant quiescent line component in both sources. This suggests that the phenomenon producing the greater velocity dispersion does not significantly heat the molecular material.