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 0.2-0.4r25. 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 discs is 6.1 ± 1.0 km s-1 (standard deviation of 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 H I 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 disc against gravitational collapse, especially in those regions of the disc which are H2 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 2, 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 to 1 and that the high star formation rates seen at z= 1 may be partly due to the presence of physically larger molecular gas discs.