An analysis is presented of the steady-state diffusion of spherical Brownian partcles through an effectively infinite expanse of fluid perforated by a circular cylindrical fiber. Detailed Stokes-flow calculations quantify the solute-fiber hydrodynamic interaction in three regimes covering the full range of solute-fiber separations, and lead to an approximate expression for the position-dependent diffusion dyadic. Thereafter, solution of the pertinent Brownian dynamic problem yields the perturbation caused by the fiber to a linear variation of the solute concentration. The results illustrate a general conclusion, derivable by simple scaling arguments, that the normal derivative of the solute concentration must tend to zero with decreasing solute-wall gap epsilon as [In(1/epsilon)](-1).