For singularly perturbed systems \(\dot x=f(t,x,z,\mu)\), \(\mu\dot z=g(t,x,z,\mu)\), \textit{A. Saberi} and \textit{H. Khalil} [IEEE Trans. Autom. Control AC--29, 542-550 (1984; Zbl 0538.93049)] have shown that if both the reduced-order system \((\mu=0)\) and the boundary-layer system are exponentially stable, then also the full-order system is stable for sufficiently small values of the perturbation parameter \(\mu\). The authors present a thorough investigation of the rate of convergence of the full-order system. They prove that, provided that some further regularity assumptions are satisfied, the rate of convergence of the full-order system approaches that of the reduced-order system as \(\mu\) approaches zero. Exponentially decaying norm bounds are given for the ``slow'' and ``fast'' components of the full-order system trajectories. To achieve this result, a new converse Lyapunov theorem for exponentially stable systems is presented.