Production of a substantially constant wall thickness in every cross section on cylindrical tubular parts produced for example by extrusion and rolling. The process of the invention comprises determining, by means of a suitable apparatus, the position of the center of the internal contour of a cross section of the tubular part, then rotating one with respect to the other a machining tool and the part about the axis of said internal contour and moving the machining tool towards the part until, over the periphery of said cross section, a constant wall thickness which is equal at most to the minimum thickness is obtained, and then repeating those operations on the successive desired sections of the tube, which are parallel to the first section. The invention also concerns the apparatus which is especially adapted for carrying out that process, the apparatus essentially comprising an ultrasonic checking apparatus (3), a machine tool (8) and an apparatus (6) for transfer from the apparatus (3) to the machine tool (8), while maintaining the orientation of the tube about its axis. The invention makes it possible to produce tubular parts with a degree of relative eccentricity in every section which is less than 1% and in particular to improve the quality of the fuel sheathing tubes produced by rolling such tubular parts.
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