We analyse and numerically study spline difference methods applied to a singularly perturbed convection-diffusion two-point boundary value problem whose solution has a single boundary layer. The method is derived by collocation with piecewise exponential splines from $C^1[0,1]$ on an regular mesh. The errors at the grid points is bounded by $Ch^4/(\varepsilon^2+h^2)$, $C$ is a constant independent of small parameter $\varepsilon$ (multiplying the highest derivative) and mesh size $h$.