It is shown that if a simple group G acts conformally on a hyperbolic surface of least area (or alternatively, on a Riemann surface of least genus σ ≥ 2), then G is normal in Aut(S) and the map Aut(S) → Aut(G) induced by conjugation is injective. For the preponderance of these minimal actions the group Aut(S)/G is isomorphic to a subgroup of Σ3. It is shown how to compute Aut(S) purely in terms of the group–theoretic structure of G, in these cases. As examples and as part of the proof, the minimal actions and the groups Aut(S) are completely worked out for A5, PSL3(3), M11 and M12.