Holomorphic principal bundles over a compact Riemann surface X that admits a flat connection are considered. A holomorphic G-bundle over X, where G is a connected semisimple linear algebraic group over ${\Bbb C}$, admits a flat connection if and only if the adjoint vector bundle admits one. More generally, for a complex reductive group G, the necessary and sufficient condition on a G-bundle to admit a flat connection is described. This simplifies the criterion obtained by the authors and given in Math. Ann. 322 (2002) 333–346.