In this paper, an exponentially fitted arithmetic average difference scheme is proposed to solve singularly perturbed differential equations with dual layer behaviour. In this method, we have extended the arithmetic average finite difference method to the second order singularly perturbed two-point boundary value problem. We have introduced a fitting factor in a three point arithmetic average discretization for the given problem which takes care of the rapid changes that occur in the boundary layers due to the perturbation parameter. This fitting factor is obtained from the asymptotic approximate solution of singular perturbations. The discrete invariant imbedding algorithm is used to solve the tridiagonal system of the fitted method. Maximum absolute errors of the several numerical examples are presented to illustrate the proposed method.