The present invention describes a hydrocracking and/or hydrotreatment process using a catalyst comprising an active phase containing at least one hydrogenating/dehydrogenating component selected from the group VIB elements and the non-precious elements of group VIII of the periodic table, used alone or in a mixture, and a support comprising at least one dealuminated zeolite Y having an overall initial atomic ratio of silicon to aluminium between 2.5 and 20, an initial weight fraction of extra-lattice aluminium atoms greater than 10%, relative to the total weight of aluminium present in the zeolite, an initial mesopore volume measured by nitrogen porosimetry greater than 0.07 ml·g−1 and an initial crystal lattice parameter a0 between 24.38 Å and 24.30 Å, said zeolite being modified by a) a stage of basic treatment comprising mixing said dealuminated zeolite Y with a basic aqueous solution, and at least one stage c) of thermal treatment.