We present a comprehensive study of the velocity dispersion of the atomic (HI) and molecular (H_2_) gas components in the disks (R{<~}R_25_) of a sample of 12 nearby spiral galaxies with moderate inclinations. Our analysis is based on sensitive high-resolution data from the THINGS (atomic gas) and HERACLES (molecular gas) surveys. To obtain reliable measurements of the velocity dispersion, we stack regions several kiloparsecs in size, after accounting for intrinsic velocity shifts due to galactic rotation and large-scale motions. We stack using various parameters: the galactocentric distance, star formation rate surface density, HI surface density, H_2_ surface density, and total gas surface density. We fit single Gaussian components to the stacked spectra and measure median velocity dispersions for HI of 11.9+/-3.1km/s and for CO of 12.0+/-3.9km/s. The CO velocity dispersions are thus, surprisingly, very similar to the corresponding ones of HI, with an average ratio of {sigma}_HI_/{sigma}_CO_=1.0+/-0.2 irrespective of the stacking parameter. The measured CO velocity dispersions are significantly higher (factor of ~2) than the traditional picture of a cold molecular gas disk associated with star formation. The high dispersion implies an additional thick molecular gas disk (possibly as thick as the HI disk). Our finding is in agreement with recent sensitive measurements in individual edge-on and face-on galaxies and points toward the general existence of a thick disk of molecular gas, in addition to the well-known thin disk in nearby spiral galaxies.