Consistent string propagation in general backgrounds requires conformal invariance. At the tree level, this requires Gell- Mann-Low β functions for the geometric couplings to vanish. We study the higher loop/hole effects on string amplitudes in a background spacetime in the context of oriented open and closed bosonic strings. We show that leading divergences can be made to cancel up to O(g⁰) (i.e., up to genus one) in string perturbations by a Fischler-Susskind-type mechanism. The consistent background depends on the gauge group chosen. This may lead to finite type-I superstring theories with gauge groups other than SO(32)