  We have discussed a particular class of exact cosmological solutions of the
4-dimensional low energy string gravity in the string frame. In the vacuum
without matter and the 2-form fields, the exact cosmological solutions always
give monotonically shrinking universes if the dilaton field is not a constant.
However, in the presence of the 2-form fields and/or the radiation-like fluid
in the string frame, the exact cosmological solutions show a minimum size of
the universe in the evolution, but with an initial cosmological curvature
singularity in the string frame.
