The dust, atomic gas, and molecular gas content of a sample of 26 of isolated, degree-sized infrared clouds are compared. Half of the clouds have an infrared excess indicating the presence of H2, and 14 contain compact CO-emitting regions. Complete, high angular resolution H I and CO maps of one cloud, G236+39, resolve the transition between atomic and molecular H, as well as the location of CO formation. Assuming the infrared emission traces the total column density, H2 is inferred to be much more widely distributed than the CO. The CO rotational levels are subthermally excited, and the (2 - 1)/(1 - 0) line ratios suggest a density n(H2) approximately 200/cm(exp 3) where CO was detected. A model of H2 formation on grain surfaces balanced by self-shielded photodissociation fits the variation of infrared brightness with H I column density. Assuming a temperature of 80 K, typical of diffuse H2 (Savage et al. 1977), the H2 chemistry requires an average density n(H + 2H2) approximately 50/cm(exp 3). For G236+39, if the distance is 100 pc, the H I and H2 masses are estimated to be 90 and 70 solar mass, respectively. High-resolution H I and infrared maps of a smaller cloud, G249.0+73.7, reveal no evidence for molecular gas, which is likely due to the low total column density through this cloud. These results suggest the H2 and H I content are comparable for some interstellar cirrus clouds with column densities N(H I) greater than 4 x 10(exp 20)/cm(exp 2), even where CO was not detected.