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 Riemann surface holds for both open and closed string amplitudes and is independent of modular invariance and supersymmetry. The mass scale for the open strings stretched between Dbranes suggests a natural world-sheet ultraviolet regulator in the string path integral, preserving both T-duality and open-closed string world-sheet duality. Note added (Jan 2005): Comments and related references added.