The 1.2 m telescope at the Center for Astrophysics was used to conduct the first sensitive and systematic search for molecular gas in the inner Galaxy more than 100 pc from the Galactic plane. With an rms sensitivity per spectrum of 0.1 K at Delta v = 1.3 km s(-1) and a sampling interval of 3'.75 (0.43 beamwidth), the survey is 3-10 times more sensitive per unit solid angle than existing CO surveys of the plane, and 16 times more sensitive than the only previous extensive wide-latitude survey. Three selected regions centered on the plane were mapped, each similar to 1 degrees wide in l and similar to 8 degrees wide in b, at Galactic longitudes 30 degrees, 40 degrees, and 50 degrees; in all three there is CO emission near the terminal velocity extending up to 3 degrees from the plane. This CO is evidence for a faint, thick molecular disk in the inner Galaxy similar to 3 times as wide as that of the dense central CO layer, and comparable in width to the central H I layer. In the inner Galaxy regions surveyed, similar to 5% of the total emission, and possibly a comparable fraction of the H-2 mass, lies above the previously studied thin molecular disk.