This paper deals with the singularly perturbed boundary value problem for the second order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. An exponentially fitted difference scheme on a uniform mesh is accomplished by the method based on cubic spline in compression. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter, which is illustrated with numerical results. © 2015, Pramod Chakravarthy et al.