An apparatus for machining a tubular piece, wherein the apparatus includes a frame element, support rollers, which are associated with the frame element, support a workpiece and rotate the same around its longitudinal axis. In addition, the frame element is provided with a machining unit adapted to be movable towards and away from the workpiece. The machining unit includes a tool member, such as for example a cutting tool, as well as support wheels adapted to bear against a workpiece. The machining unit is arranged to link up with the frame element through the use of an auxiliary frame element adapted to clamp itself thereto at not less than two support points, for example through the use of a self wedging assembly implemented with an indentation.