  We examine the string cosmology equations with a dilaton potential in the
context of the Pre-Big Bang Scenario with the desired scale factor duality, and
give a generic algorithm for obtaining solutions with appropriate evolutionary
properties. This enables us to find pre-big bang type solutions with suitable
dilaton behaviour that are regular at $t=0$, thereby solving the graceful exit
problem. However to avoid fine tuning of initial data, an `exotic' equation of
state is needed that relates the fluid properties to the dilaton field. We
discuss why such an equation of state should be required for reliable dilaton
behaviour at late times.
