A composite material part having a matrix made of ceramic, at least for the most part, is fabricated by a method including making a fiber preform from silicon carbide fibers containing less than 1 at % oxygen; depositing a boron nitride interphase on the fibers of the preform, deposition being performed by chemical vapor infiltration at a deposition rate of less than 0.3 μm/h; performing heat treatment to stabilize the boron nitride of the interphase, after the interphase has been deposited, without prior exposure of the interphase to an oxidizing atmosphere and before depositing matrix layer on the interphase, the heat treatment being performed at a temperature higher than 1300° C. and not less than the maximum temperature to be encountered subsequently until the densification of the preform with the matrix has been completed; and thereafter, densifying the perform with a matrix that is made of ceramic, at least for the most part.