  Let U be an open subset of a unirational variety (or more generally of a
separably rationally connected variety). We prove that there is rational curve
C in U such that the fundamental group of C surjects onto the fundamental group
of U. The proof improves the earlier proof (math.AG/0003138) and works in any
characteristic. Much improved version, thanks to comments of
Colliot-Th\'el\`ene.
