We present extensive experimental data on charge transport by Andreev reflection through a mesoscopic superconducting island coupled to two normal metal leads through small tunnel junctions and to a capacitive gate. The data show the systematic dependence on the gate-induced charge Q0 of both the Coulomb blockade of Andreev reflection and of striking voltage thresholds that correspond to specific tunneling processes. The magnitude of this Andreev current and its dependence on Q0 are in excellent agreement with recent theory, provided that interference between the two tunneling electrons is taken into account.