A process and apparatus for electrically machining an electrode workpiece by means of an electrode tool, in which the electrode workpiece is first machined by EDM by way of electrical discharges occuring across a dielectric fluid. The electrode workpiece is subsequently subjected to a finishing pass in the course of which it is machined by electrochemical machining (ECM) through a conductive electrolyte replacing the dielectric fluid. The machining phase by ECM is effected by means of the same electrode tool or by means of an electrode tool of the same dimensions as is used during the machining phase with EDM. During machining by EDM, the electrode tool is orbited, or translated, in a direction perpendicular to the axis of feed of the electrode tool relative to the electrode workpiece, with an amplitude larger than the amplitude during machining by ECM.