Method for the production of mechanical or chemi-mechanical pulp as raw material for paper or cardboard. According to this method, the pulp is fibrillated and the fibrillated pulp is bleached in alkaline conditions. According to the present invention, the pulp is screened to separate the reject from the accept, at maximum approximately 60% of the total amount of pulp is separated as the reject, the reject is bleached separate from the accept, and, after that, the bleached reject is remixed with the accept. When operating according to the present invention, the strength of the pulp increases and the energy used for refining is reduced, which is seen both in the refining of the reject and in the post-refining of the final mechanical pulp.