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In this study, numerical solution of a differential-difference equation with a boundary layer at one end of the domain is suggested using an exponential spline. The numerical scheme is developed using an exponential spline with a special type of mesh. A fitting parameter is inserted in the scheme to improve the accuracy and to control the oscillations in the solution due to large delay. Convergence of the method is examined. The error profiles are represented by tabulating the maximum absolute errors in the solution. Graphs are being used to show that how the fitting parameter influence the layer structure.