The diffusion controlled growth of a compound phase AnB between two thin films of material A and B is studied with the nonlinear Kirkendall effect included. This growth process is important in electronic materials processing and in synthesis of high-temperature materials using multilayer films. Previous models of the growth rate do not solve the diffusion equation, and thus do not fully utilize the predictive capability. This paper describes a self-similar transformation that reduces the nonlinear, time-dependent diffusion equation with two free boundaries into a nonlinear ordinary differential equation, which is solved numerically by a shooting method. It is found that the intrinsic diffusion coefficients of A and B in AnB can be determined from the positions of the interfaces without using the concentration profile. This provides a simpler method for measuring intrinsic diffusion coefficients. An asymptotic solution valid for small concentration gradients is derived and agrees with the numerical results.