A singularly perturbed two-point boundary value problem with an exponential boundary layer is solved numerically by using an adaptive grid method. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the exact solution. The error analysis for this approach was carried out by Qiu et al. (J. Comput. Appl. Math. 101 (1999) 1-25). In this work, their error bound will be improved to the optimal order which is independent of the perturbation parameter. The main ingredient used to obtain the improved result is the theory of the discrete Green's function.