The diffusion coefficient D(C) for a very dilute aqueous electrolyte is represented, in a first approximation, by D(C) = D(0)(1 - k rootC), where C is the electrolyte concentration and k is a positive parameter. For the experimental setup in which, initially, a sharp boundary is formed between an electrolyte solution and pure water, the Gosting-Fujita method is not applicable for solving free diffusion with this concentration-dependent D, because D is not regular at C = 0. In this paper, another perturbation method which uses k as a perturbation parameter is applied to solve this diffusion problem correct up to second order in k, and the apparent diffusion coefficient D-j defined in the Rayleigh fringe method for measuring transport processes in solutions is calculated as a function of the position in the diffusion boundary. It is shown that the result does not support Albright and Miller's finding from numerical solutions that the position at which D-j agrees with (D) over bar is independent of k. Here, (D) over bar is the D value at the mean concentration.