A thermostructural composite material part including carbon or ceramic fiber reinforcement densified by a matrix having at least one thin portion in which: the thickness of the part is less than 2 mm, or indeed less than 1 mm; the fiber reinforcement is made as a single thickness of multilayer fabric made of spread yarns having a weight of not less than 200 tex; the fiber volume ratio lies in the range 25% to 45%; and the ratio between the number of layers of the multilayer fabric and the thickness in millimeters of the part is not less than four.