The invention relates to a distillation apparatus (1) which is suitable in particular for the distillation of sea water into fresh water. The apparatus comprises a plurality of flat, bag-like elements (3) formed from a thin film material such as plastic film and placed one against the other, the elements serving as heat exchangers between a vaporizing liquid flowing along the exterior surfaces of the element and a condensing vapor directed to the inside of the element, and a compressor (15) for increasing the pressure and temperature of the generated vapor before it is directed to the inside of the elements. The essential idea of the invention is that at the upper end of each bag-like element (3) there is a honeycomb-structured end strip (4) having substantially the width of the element, the strip containing parallel feeding ducts (26) separated from each other by partition walls, the ducts distributing the liquid to be evaporated over the entire width of the element surface. In addition, the strip (4) may contain ducts (25), formed in the same manner, which direct the vapor to be condensed to the inside of the element over the entire width of its end. At the lower end of the element there may additionally be a honeycomb-structured end strip (7), the ducts contained therein collecting the produced distillate into a collection vessel (8) and discharging any unvaporized liquid into a collection basin (9).