We present a comprehensive study of the velocity dispersion of the atomic ( H i ) and molecular ( H _ { 2 } ) gas components in the disks ( R \lesssim 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 kilo-parsecs in size , after accounting for intrinsic velocity shifts due to galactic rotation and large–scale motions . We stack using various parameters : the galacto-centric distance , star formation rate surface density , H i 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 H i of 11.9 \pm 3.1 km s ^ { -1 } and for H _ { 2 } of 12.0 \pm 3.9 km s ^ { -1 } . The CO velocity dispersions are thus , surprisingly , very similar to the corresponding ones of H i , with an average ratio of \sigma _ { HI } / \sigma _ { CO } = 1.0 \pm 0.2 irrespective of the stacking parameter . The measured CO velocity dispersions are significantly higher ( factor \sim 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 H i disk ) . Our finding is in agreement with recent sensitive measurements in individual edge–on and face–on galaxies and points towards the general existence of a thick disk of molecular gas , in addition to the well–known thin disk in nearby spiral galaxies .