A fiber structure for forming the reinforcement of a part to be made is placed in a mold (22) having at least one wall formed by a flexible membrane, and then a resin composition is injected into the mold, the composition having a volatile material content of less than 25% by weight and being at a temperature of a value such that its viscosity lies in the range 0.1 Pa·s to 0.3 Pa·s. The resin is polymerized in the mold placed in an enclosure (20) with temperature being raised progressively and with polymerization including at least a final stage of polymerization under pressure in order to obtain a composite material part presenting residual porosity of less than 11% by volume.