We have constructed three-dimensional models for the equilibrium abundance of molecular hydrogen within diffuse interstellar clouds of arbitrary geometry that are illuminated by ultraviolet radiation. The position-dependent photo- dissociation rate of H$_2$ within such clouds was computed using a 26-ray approximation to model the attenuation of the incident ultraviolet radiation field by dust and by H$_2$ line absorption. We have applied our modeling technique to the isolated diffuse cloud G236+39, assuming that the cloud has a constant density and that the thickness of the cloud along the line of sight is at every point proportional to the 100 um continuum intensity measured by IRAS. We find that our model can successfully account for observed variations in the ratio of 100 umu continuum intensity to HI column density, with larger values of that ratio occurring along lines of sight in which the molecular hydrogen fraction is expected to be largest. Using a standard chi^2 analysis to assess the goodness of fit of our models, we find (at the 60sigma level) that a three-dimensional model is more successful in matching the observational data than a one-dimensional model in which the geometrical extent of the cloud along the line of sight is assumed to be very much smaller than its extent in the plane-of-the-sky. If D is the distance to G236+39, and given standard assumptions about the rate of grain-catalysed H_2 formation, we find that the cloud has an extent along the line of sight that is 0.9+-0.1 times its mean extent projected onto the plane of the sky; a gas density of (53+-8)(100 pc/D) cm^-3; and is illuminated by a radiation field of (1.1+-0.2) (100 pc/D) times the mean interstellar radiation field estimated by Draine (1978). The derived 100 um emissivity per nucleon is (1.13+-0.06)x10^-20 MJy sr^-1 cm^2.

27 pages LaTex, uses aaspp4.sty, ApJ August 1