  We examine the effect of spatial curvature in the pre-big-bang inflationary
model suggested by string theory. We study $O(\alpha ')$ corrections and we
show that, independently of the initial curvature, they lead to a phase of
exponential inflation. The amount of inflation in this phase is long enough to
solve the horizon and flatness problems if the evolution starts deeply into the
weak coupling regime. There is a region of the parameter space of the model
where such a long inflationary phase at the string scale is consistent with
COBE anisotropies, millisecond pulsar timing and nucleosynthesis constraints.
We discuss implications for the spectrum of relic gravitational waves at the
frequencies of LIGO and Virgo.
