Analytic solutions to the steady state radial diffusion equation allowing for a distributed source and sink of ions are presented. The diffusion space has absorption boundaries at the ends and is divided into two regions separated by an interior boundary where a discontinuity in diffusion and loss parameters is allowed to exist. The most general solution consists of a collection of point sources with adjustable weights to approximate an arbitrary continuous distribution of ion sources. The theory is applied to the study of thermal ion diffusion in Saturn's magnetosphere. It is concluded that a relatively large transport rate is required to conform with low-energy Voyager plasma measurements and a radial diffusion coefficient expressible as DLL=4¿10-7 s-1 (L/6)3--4 is indicated. In such a fast diffusion regime cool O+ ion densities at L=2.8, 6, and 15 may be obtained from a single source in the form of the Dione-Tethys neutral H2O cloud. Hot ion densities in the outer magnetosphere are obtained from ionization of Titan's neutral atomic clouds of hydrogen and nitrogen and ion pickup by the magnetospheric flow. The O+ thermal ion model is compatible with a bimodal distribution of neutral hydrogen in the magnetosphere with a Titan cloud [H>≂20 cm-3 (L/20) in the outer magnetosphere and a low-density [H>=6 cm-3 (6/L)2 hydrogen corona of either Saturnian origin in the inner magnetosphere. ¿American Geophysical Union 1990