A method of fabricating a composite material part having a body secured to at least one platform present at one end of the body, the method including using the fiber blank to form a preform for the part that is to be made by folding out segments of the second and third portions that are not interlinked with the first portion on opposite sides of the first portion, by shaping the folded-out segments of the second and third portions to form preform portions for a platform of the part to be fabricated, and by folding down the segment of the first portion in order to fill in the groove present in the gap between the second and third portions; and densifying the preform with a matrix in order to obtain a part made of composite material having at least one integrated platform.