A method of fabricating a composite material part, the method including making a consolidated fiber preform, the fibers of the preform being carbon or ceramic fibers and being coated with an interphase; obtaining a consolidated and partially densified fiber preform, the partial densification comprising using chemical vapor infiltration to form a first matrix phase on the interphase; and continuing densification of the fiber preform by infiltrating an infiltration composition containing at least silicon and at least one other element suitable for lowering the melting temperature of the infiltration composition to a temperature less than or equal to 1150° C.