A microfluidic network having at least one channel which is open on the top side and in which one end of at least one hollow fiber is inserted and a process for making it. For this it is provided that the outside dimensions of the hollow fiber are matched to the inside dimensions of the channel, capillary channels being formed between the hollow fiber and the walls of the channel. The network is covered with a cover film or plate which closes the channel and preferably fixes the hollow fiber temporarily in the channel, and the capillary channels are filled with a fluidically sealing cement. In the area of the inserted end of the hollow fiber, a capillary stop structure is formed so that the cement as it is added it cannot seal the inserted open end of the hollow fiber.