The entropic depletion force, in colloids, arises when large particles are placed in a solution of smaller ones, and sterically constrained to avoid them. In this paper, we consider a system of two parallel plates suspended in a semidilute solution of long thin rods of length L and diameter D. By numerically solving an integral equation, which is exact in the "Onsager Limit" (D much less than L), we obtain the depletion force between the plates. The second integral of this determines (via the Derjaguin approximation) the depletion potential between two large hard spheres of radius R, immersed in a solution of hard rods (satisfying D much less than L less than or equal to R). The results for this potential are compared with our previous second order perturbation treatment [Y. Mao, M. E. Cates, and H. N. W. Lekkerkerker, Phys. Rev. Lett. 75, 4548 (1995)], as well as with newly computed third order perturbation results. There is good agreement at low and intermediate densities (which validates our numerical procedures for the integral equation) but the gradual failure of the perturbative treatments is revealed as the density increases. From the numerical results, we conclude that for typical colloidal sphere/rod mixtures, the repulsive barrier in the depletion interaction is Likely to be less than the thermal energy k(B)T throughout the semidilute concentration range of the rods. This reverses our previous conclusion, based on second order perturbation theory, that this barrier could easily lead to kinetic stabilization of the mixture. However, it is possible that terms of order D/L (neglected here) could further modify the results. Within the present theory, the main modification to the well-known (attractive) first order interaction, as the density is raised, is by changing the depth and range of the primary attractive well.