  In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the
round 3-sphere of radius 1) we classify the stable constant mean curvature
spheres, showing that in some Berger spheres (a close to 0) there are unstable
constant mean curvature spheres. Also, we classify the orientable compact
stable constant mean curvature surfaces in S^3(a), 1/3 <= a < 1 proving that
they are spheres or the minimal Clifford torus in S^3(1/3). This allows to
solve the isoperimetric problem in these Berger spheres.
