An approximate solution of the Milne problem in a finite slab for isotropic elastic scattering in a non-absorbing medium is investigated. The problem may be formulated as an integral equation of the Wiener-Hopf type with a finite range of integration. A solution to this integral equation is obtained by the use of finite Fourier transforms, which lead to an infinite system of linear algebraic equations. A highly convergent iterative solution for the truncated system of equations is developed. Numerical results are presented for several slab thicknesses in the case of normal illumination.