Ultraviolet limit of open string theory

We confirm the intuition that a string theory which is perturbatively infrared finite is automatically perturbatively ultraviolet finite. Our derivation based on the asymptotics of the Selberg trace formula for the Greens function on a riemannian surface holds for both open and closed string amplitudes and is independent of modular invariance and supersymmetry. The masses of the open strings stretched between D-branes give a natural world-sheet ultraviolet regulator for the string path integral preserving both T-duality and open-closed string world-sheet duality.