Abstract: Membrane module for substance-specific treatment of fluids, such that the module has first and second membrane elements arranged with one end pointing toward a distribution space and the other end pointing toward a collection space. Each of the membrane elements have a cavity formed by a membrane wall. The first membrane elements have their ends embedded in sealing compounds such that their cavities open into the distribution space and collection space. The second membrane elements are also embedded in the sealing compound at the end pointing toward the collection space. The cavities of the second membrane elements open into the collection space, but are closed at the end pointing toward the distribution space. The fluid to be treated flows through the first membrane elements in cross-flow mode. In the process, part of the fluid passes through the membrane wall as a permeate, which then passes through the second membrane elements in dead-end mode.