Graceful exit in quantum string cosmology

We write an O(d,d)-covariant Wheeler-De Witt equation in the (d^2+1)-dimensional minisuperspace of low-energy cosmological string backgrounds. We discuss explicit examples of transitions between two duality-related cosmological phases, and we find a finite quantum transition probability even when the two phases are classically separated by a curvature singularity. This quantum approach is completely free from operator ordering ambiguities as a consequence of the duality symmetries of the string effective action.