  We give a full classification of complete rotationally invariant surfaces
with constant Gauss curvature in Berger spheres: they are either Clifford tori,
which are flat, or spheres of Gauss curvature $K \geq K_0$ for a positive
constant $K_0$, which we determine explicitly and depends on the geometry of
the ambient Berger sphere. For values of $K_0 \leq K \leq K_P$, for a specific
constant $K_P$, it was not known until now whether complete constant Gauss
curvature $K$ surfaces existed in Berger spheres, so our classification
provides the first examples. For $K > K_P$, we prove that the rotationally
invariant spheres from our classification are the only topological spheres with
constant Gauss curvature in Berger spheres.
