An analysis of large-area CO J =3-2 maps from the James Clerk Maxwell Telescope for 12 nearby spiral galaxies reveals low velocity dispersions in the molecular component of the interstellar medium . The three lowest luminosity galaxies show a relatively flat velocity dispersion as a function of radius while the remaining nine galaxies show a central peak with a radial fall-off within 0.2 - 0.4 r _ { 25 } . Correcting for the average contribution due to the internal velocity dispersions of a population of giant molecular clouds , the average cloud-cloud velocity dispersion across the galactic disks is 6.1 \pm 1.0 km s ^ { -1 } ( standard deviation 2.9 km s ^ { -1 } ) , in reasonable agreement with previous measurements for the Galaxy and M33 . The cloud-cloud velocity dispersion derived from the CO data is on average two times smaller than the HI velocity dispersion measured in the same galaxies . The low cloud-cloud velocity dispersion implies that the molecular gas is the critical component determining the stability of the galactic disk against gravitational collapse , especially in those regions of the disk which are H _ { 2 } dominated . The cloud-cloud velocity dispersion shows a significant positive correlation with both the far-infrared luminosity , which traces the star formation activity , and the K-band absolute magnitude , which traces the total stellar mass . For three galaxies in the Virgo cluster , smoothing the data to a resolution of 4.5 kpc ( to match the typical resolution of high redshift CO observations ) increases the measured velocity dispersion by roughly a factor of two , comparable to the dispersion measured recently in a normal galaxy at z = 1 . This comparison suggests that the mass and star formation rate surface densities may be similar in galaxies from z = 0 - 1 and that the high star formation rates seen at z = 1 may be partly due to the presence of physically larger molecular gas disks .