Abstract: The following process steps are carried out in order to purify a liquid by membrane distillation, in particular to produce desalinated water from seawater or brackish water or process water: passing a relatively warn vaporising stream of liquid (retentate stream) over a porous membrane, vapour flowing via the pores of the membrane to the other side of said membrane, and condensing said vapour on a relatively cool condenser surface to give a distillate stream, said condenser surface forming the non-porous separation between a feed stream to be purified and said distillate steam, which feed stream is in counter-current with the retentate stream so that an appreciable proportion of the latent heat will be transferred via vapour to the feed stream, and a gas gap with a width of less than 5 mm being present between the porous membrane and the condenser surface.