Analyses of the far-IR continuum emission from dust and the gamma - ray emission from cosmic-ray interactions with gas have recently revealed the presence of extensive clouds of ``dark gas'' in the Galaxy. This dark gas is essentially invisible in 21-cm HI and 3-mm CO emission. It surrounds all nearby dense molecular clouds and appears to form a ``bridge'' to the larger-scale distribution of atomic gas. The spatial distribution of this dark gas has much in common with the ubiquitous faint OH emission in the Galaxy discovered recently in a ``blind'' mini-survey with the Onsala radio telescope. I will summarize a new "pilot" program we have just completed with the GBT to begin a study of the spatial structure of the Galactic OH emission and its possible relation to the dark gas, and explain why the GBT is particularly well suited to this problem.