  We study Big Crunch/Big Bang cosmologies that correspond to exact world-sheet
superconformal field theories of type II strings. The string theory spacetime
contains a Big Crunch and a Big Bang cosmology, as well as additional
``whisker'' asymptotic and intermediate regions. Within the context of free
string theory, we compute, unambiguously, the scalar fluctuation spectrum in
all regions of spacetime. Generically, the Big Crunch fluctuation spectrum is
altered while passing through the bounce singularity. The change in the
spectrum is characterized by a function $\Delta$, which is momentum and
time-dependent. We compute $\Delta$ explicitly and demonstrate that it arises
from the whisker regions. The whiskers are also shown to lead to
``entanglement'' entropy in the Big Bang region. Finally, in the Milne orbifold
limit of our superconformal vacua, we show that $\Delta\to 1$ and, hence, the
fluctuation spectrum is unaltered by the Big Crunch/Big Bang singularity. We
comment on, but do not attempt to resolve, subtleties related to gravitational
backreaction and light winding modes when interactions are taken into account.
