  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.
