  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.
