  We consider groups of orientation-preserving real analytic diffeomorphisms of
the circle which have a finite image under the rotation number function. We
show that if such a group is nondiscrete with respect to the $C^1$-topology
then it has a finite orbit. As a corollary, we show that if such a group has no
finite orbit then each of its subgroups contains either a cyclic group of
finite index or a nonabelian free subgroup.
