A method of fabricating a composite material part such as a link. The method comprises the successive operations of applying reinforcing fiber layers (14a, 14b, 14c) that are braided around a mandrel over all or part of the length of the mandrel, the layers being superposed one on another. The method includes a step of cutting an intermediate reinforcing fiber layer (14a, 14b, 14c) after it has been applied, a step of withdrawing a cut portion prior to applying the following braided layer (14b, 14c), thereby locally building up an extra thickness of reinforcing layers on each portion of the cut layer that has remained in place, the extra thickness constituting local reinforcement of the part. Resin is subsequently injected into the various braided layers prior to the resin being polymerized in order to prepare a blank. The method applies in particular to the field of aviation.