Method for continuous cleaning of process water circulating in a device for processing waste paper (100) and/or in a paper machine (200) of a machine for manufacturing paper from waste paper, in which process water to be cleaned is supplied, from at least one unit of the device (100) and/or the paper machine (200), to a process water processing unit having an anaerobic reactor (125a, 125b, 225) comprising one or more gas separators (504), the process water is contacted, in the anaerobic reactor (125a, 125b, 225), with anaerobic microorganisms in order to break down impurities contained in the process water, and the cleaned process water is conveyed away from the anaerobic reactor (125a, 125b, 225) and returned to at least one unit of the device (100) and/or the paper machine (200), wherein at least one unit is selected from one or more of a pulper (105), sorting device (110a, 110b), dewatering unit (115a, 115b), oxidation/reduction unit (112), centrifugal sorter (202), fine sorting device (204), paper-machine mold part (206), paper-machine press part (208), reject treatment unit (118a, 118b, 212), fiber recovery unit (214) and drying part (210), wherein, in the method, a content of inorganic solids in the process water contained in the anaerobic reactor (125a, 125b, 225) is measured continuously or discontinuously and, when the content of inorganic solids in the process water contained in the anaerobic reactor (125a, 125b, 225) exceeds a pre-set limit value, process water is continuously supplied, from the anaerobic reactor (125a, 125b, 225), to a solid-liquid separator (126a, 126b, 226) for separation into a fraction depleted of inorganic solids and a fraction enriched with inorganic solids, wherein the fraction depleted of inorganic solids is returned to the anaerobic reactor (125a, 125b, 225) at least in part and the fraction enriched with inorganic solids is conveyed away from the device (100) and/or the paper machine (200), until the content of inorganic solids in the process water contained in the anaerobic reactor (125a, 125b, 225) is less than the pre-set limit value.