The J = 2-1 transition of CO has been detected from four positions in NGC 185; upper limits are placed on emission from an additional four positions, which include the peak of the atomic gas distribution. Using what we believe to be an appropriate conversion factor from CO integrated intensity to H_2_ column density yields a minimum mass ratio M(H_2_)/M(H I) of 4 for NGC 185. Large velocity gradient (LVG) models, weakly constrained by the observed intensity ratio I_2-1_/I_1-0_ and the argument that T_k_ <= 10 K, yield number densities in excess of several times 10^2^ for the molecular gas. Our measurements clearly confirm the large-scale offset between the atomic and molecular phases in NGC 185. We suggest that this results from the stochastic nature of the gas recycling process in a dwarf spheroid. Molecular gas is associated with the two optical dust clouds but is also found within the young star population in which dust has not been detected. From recent optical photometry of the young stars, we calculate the mean intensity of interstellar UV radiation important for CO photodissociation, finding values only a few percent of that typical near the Sun. Under these conditions, molecular self- shielding is likely to be much more important than dust shielding for the survival of molecules in the interstellar medium, making optical dust clouds unreliable tracers of molecular gas in this and similar galaxies. Presently available observations of the ISM and stellar populations in NGC 185 are consistent with two scenarios for gas recycling over long timescales. We discuss these scenarios and their implications for the frequency of star formation activity in dwarf elliptical galaxies.