Traditional measurements of molecular gas via rotational and hyperfine transitions of diatomic species such as CO, CH, and OH are only able to detect the cores and adjacent low-density molecular regions of interstellar molecular clouds. Longer than usual integration times of the CO(J = 1-0) line at 115 GHz and the OH 1667 MHz ground state, hyperfine, main line transition have revealed the presence of low-density, diffuse gas surrounding the molecular core MBM40. Using the Schlegel, Finkbeiner and Davis (1998) dust maps in conjunction with new CO and OH observations, the amount of molecular gas in the diffuse outer regions of MBM40 has been determined to be equal if not greater than the mass in the core region. This much mass was not expected to be found in the outer most regions of the cloud. In addition, a re-analysis of high-latitude CO survey data reveals that this low-density molecular gas may be widespread in the Galaxy. The relationship between the low-density, diffuse molecular component studied in this thesis and the recently identified ``dark' molecular gas is discussed.