An installation for shaping a fiber preform as a body of revolution presenting a profile that varies in radial section, the installation including a storage mandrel for storing a fiber texture, follower rollers, and a mold in the form of a body of revolution on which the fiber texture is to be shaped by winding, the follower roller(s) being placed between the storage mandrel and the mold in the form of a body of revolution. The storage mandrel, each follower roller, and the mold present radii across their axial widths that vary to define outer surfaces having profiles in relief. A follower roller has sectors releasably fastened on the outer surface of the roller, each sector extending over a fraction of the circumference of the roller and over all or part of the axial width of the roller.