  We consider finitely generated groups of real-analytic circle
diffeomorphisms. We show that if such a group admits an exceptional minimal set
(i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero;
moreover, there are only finitely many orbits of connected components of its
complement. For the case of minimal actions, we show that if the underlying
group is (algebraically) free, then the action is ergodic with respect to the
Lebesgue measure. This provides first answers to questions due to \'E. Ghys, G.
Hector and D. Sullivan.
