The present invention describes a method for hydrocracking and/or hydrotreating hydrocarbon-containing feeds using a catalyst comprising at least one hydro-dehydrogenizing metal selected from the group made up of group VIB and non-noble group VIII metals of the periodic table and a support comprising at least one zeolite having at least pore openings containing 12 oxygen atoms, modified by a) at least a stage of introducing at least one alkaline cation belonging to groups IA or IIA of the periodic table, b) a stage of treating said zeolite in the presence of at least one molecular compound containing at least one silicon atom, c) at least one stage of partial exchange of said alkaline cations by NH4+ cations in such a way that the proportion of alkaline cations remaining in the modified zeolite at the end of stage c) is such that the alkaline cation/aluminum molar ratio ranges between 0.2:1 and 0.01:1, and d) at least one thermal treatment stage.