  It is shown -- using a FRW model with ${\bf S}^3 \times {\bf S}^6$ as spatial
sections and a positive cosmological constant -- that classical signature
change implies a new compactification mechanism. The internal scale factor is
of the order $\Lambda^{-1/2}$, and the solutions are stable against small
perturbatons. In the case of compactified ${\bf S}^6$, it is shown that the
effective four-dimensional space-time metric has Lorentzian signature,
undergoes exponential inflation in ${\bf S}^3$ and is unique. Speculations
concerning relations to quantum cosmology and conceivable modifications are
added.
