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A model for coupled radiation transfer and thermal diffusion is proposed, which provides a local temperature field. Single-line scanning of a laser beam over a thin layer of metallic powder placed on a dense substrate of the same material is studied. Both the laser beam diameter and the layer thickness are about 50 μm. The typical scanning velocity is in the range of 10–20 cm/s. An effective volumetric heat source is estimated from laser radiation scattering and absorption in a powder layer. A strong difference in thermal conductivity between the powder bed and dense material is taken into account. The above conditions correspond to the technology of selective laser melting that is applied to build objects of complicated shape from metallic powder. Complete remelting of the powder in the scanned zone and its good adhesion to the substrate ensure fabrication of functional parts with mechanical properties close to the ones of the wrought material. Experiments with single-line melting indicate that an interval of scanning velocities exists, where the remelted tracks are uniform. The tracks become “broken” if the scanning velocity is outside this interval. This is extremely undesirable and referred to as the “balling” effect. The size and the shape of the melt pool and the surface of the metallurgical contact of the remelted material to the substrate are analyzed in relation to the scanning velocity. The modeling results are compared with experimental observation of laser tracks. The experimentally found balling effect at scanning velocities above ∼20 cm/s can be explained by the Plateau–Rayleigh capillary instability of the melt pool. Two factors destabilize the process with increasing the scanning velocity: increasing the length-to-width ratio of the melt pool and decreasing the width of its contact with the substrate.