The method is for bleaching cellulose pulp in a bleaching line that has at least two bleaching steps including a first and second bleaching steps. The bleaching steps have wash apparatuses for the pulp arranged after the first and second bleaching steps. The dilution liquor is led in counter-currently to the pulp flow through the bleaching steps in the bleach line. A substantial portion of the wash liquor is supplied by means of a main conduit so that each of the wash apparatuses is independently supplied by means of first branch conduits connected to the main conduit. Each of the wash apparatuses independently bleeds out wash filtrate by means of second branch conduits connected to the main conduit. The main conduit has an outlet end which bleeds out at least some of the wash filtrate from the wash apparatus.