  We study string propagation in a spacetime with positive cosmological
constant, which includes a circle whose radius approaches a finite value as
|t|\to\infty, and goes to zero at t=0. Near this cosmological singularity, the
spacetime looks like R^{1,1}/Z. In string theory, this spacetime must be
extended by including four additional regions, two of which are compact. The
other two introduce new asymptotic regions, corresponding to early and late
times, respectively. States of quantum fields in this spacetime are defined in
the tensor product of the two Hilbert spaces corresponding to the early time
asymptotic regions, and the S-matrix describes the evolution of such states to
states in the tensor product of the two late time asymptotic regions. We show
that string theory provides a unique continuation of wavefunctions past the
cosmological singularities, and allows one to compute the S-matrix. The
incoming vacuum evolves into an outgoing state with particles. We also discuss
instabilities of asymptotically timelike linear dilaton spacetimes, and the
question of holography in such spaces. Finally, we briefly comment on the
relation of our results to recent discussions of de Sitter space.
