Disclosed is an ash-melting furnace which can respond to fluctuations in the load and is capable of highly efficient and stable operation. In an ash-melting furnace which heats and melts primary ash (ash containing rough particles) and fly ash (ash containing minute particles) obtained by combustion together, the primary ash is supplied from an upper end of the furnace to form an upper layer of a two-tiered layer of ash, and the fly ash is supplied from the upper end of the furnace to form a lower layer of a two-tiered layer of ash. The two-tiered layer together is moved towards the far end of the furnace, and is heated and melted by a burner (10) to form molten slugs (25a) during the moving step. When the burner is an oxygen-enriched burner, this configuration makes it possible to control the volume of oxygen to be added to the burner air (including changing the density of the oxygen). This invention can control not only the volume of fuel but also the density of the oxygen-enriched air and the volume of the ash to be supplied in response to the state of flow of the molten ash melted by the oxygen-enriched burner.