  The pre--big bang cosmological scenario is studied within the context of the
Brans--Dicke theory of gravity. An epoch of superinflationary expansion may
occur in the pre--big bang phase of the Universe's history in a certain region
of parameter space. Two models are considered that contain a cosmological
constant in the gravitational and matter sectors of the theory, respectively.
Classical pre-- and post--big bang solutions are found for both models. The
existence of a curvature singularity forbids a classical transition between the
two branches. On the other hand, a quantum cosmological approach based on the
tunneling boundary condition results in a non--zero transition probability. The
transition may be interpreted as a spatial reflection of the wavefunction in
minisuperspace.
