We analyze the quantum cosmology of one-loop string effective models which exhibit an O ( d , d ) symmetry. It is shown that due to the large symmetry of these models the Wheeler-de Witt equation can completely be solved. As a result, we find a basis of solutions with well-defined transformation properties under O ( d , d ) and under scale factor duality in particular. The general results are explicitly applied to 2-dimensional target spaces while some aspects of higher dimensional cases are also discussed. Moreover, a semiclassical wave function for the 2-dimensional black hole is constructed as a superposition of our basis.