method hereof is particularly suitable for remelting there is no regulation of the welding power on account of the temperature of the molten bath surface in the region of the focus of the welding beam 16 fig3 instead a simulation of the molten bath depth is carried out a priori on the basis of heat conduction calculations along the welding path 13 and welding direction 19 the aim of adapting the welding power p is preferably to achieve a constant thickness of the molten or applied region 13 this is shown in fig1 in which a certain welding power p is present at the start 8 in the thicker region d u2032 this welding power is already being reduced 9 when the welding beam 16 fig3 approaches the thinner d u2033 region 4 or has not yet reached it the welding power p reaches a minimal value 15 only when the region 4 has already been reached by the welding beam 16 the welding power p also increases further after the thinner d u2033 region 4 has been crossed until it reaches a higher value 7 again the wall thickness d u2032 d u2033 is the thickness in the direction of a welding beam 16 fig3 or in the direction of build up of the material during the build up welding but not parallel to the surface 14 the curve profile of the welding power p is shown only schematically it is significant that the welding power p is not configured in a manner corresponding to the contour of the region 4 i e it is not reduced in a ramp like manner but instead is lowered in advance before the.