The invention concerns a process for the machining of workpieces (10) using a laser beam (11) under protective gas, which has a vapourizing effect on the material, where an electric arc out of the laser beam (11) is guided in the proximity of the workpiece processing area (12) and whose ignition is controlled by a control device (13). In order to create a contact-less ignition process that thus avoids any additional movements on individual NC axes and presents the exact synchronization options of the electric arc ignition, of the laser beam effect and of the workpiece movements, when feedback that is harmful to machine control is avoided in the latter, the process is so effected that before the beginning of the machining in the processing area (12), a laser-induced plasma (14) is produced that is extinguished by the control device (13) after the ignition of the electric arc.