In a multi-effect falling-film evaporator, purified water for special purposes is produced. In each effect, a steam phase and a water phase is produced from the feed water. The steam phase is used for heating in the subsequent effect, whereby it condenses to product water, and the water phase becomes the feed stream for the subsequent effect. A reject stream is removed from the process to carry off impurities separated from the water. According to the invention, a reject stream is removed from each effect. Preferably, each reject stream is a condensed fraction of the steam generated in the respective effect.