A general theory is presented for calculating the effective diffusivity (D) over bar of an arbitrarily shaped Brownian solute diffusing within a straight pore of arbitrary cross section. Hydrodynamic interactions with the walls as well as rotational diffusion of the solute are rigorously accounted for. Asymptotic analysis leads to the conclusion that D is given by a formula of the Brenner-Gaydos (1977) form (D) over bar/D-bulk = 1 + C-0 lambda ln lambda + C-1 lambda + 0(lambda) in the limit lambda << 1, irrespective of the particular geometry considered, where lambda denotes the ratio of solute to pore sizes. The O(lambda ln lambda) term derives from far-field hydrodynamic interactions between the solute and the local tangent to the pore wall and is universally quantified in terms of the ratio of the Stokes-Einstein equivalent radius of the solute to the hydraulic radius of the pore. The O(lambda) term depends in a complicated way upon the pore and solute shapes, the reflected field due to a Stokeslet within the pore, and hydrodynamic interactions between the solute and a plane wall. Numerical evaluation of the O(lambda) coefficient requires solution of a planar-wall translational-rotational diffusion problem. Application of the general theory is demonstrated via explicit derivation of the asymptotic formula giving (D) over bar for a dumbbell-shaped solute diffusing within a circular cylindrical pore.