A method to cold-form a semi-finished metal tubular element to produce an exhaust terminal for burnt gas evacuation systems of internal-combustion engines, said tubular element comprising an inner end suitable for being coupled with said gas exhaust system and an outer end opposite to said inner end, characterized in that it comprises the following operating steps: a. reducing the diameter of an extreme section of said tubular element to form the inner end of the exhaust terminal; b. increasing the diameter of the opposite extreme section of said tubular element to form the outer end of the exhaust terminal, so as to determine an annulus which forms an annular wall joining said outer end to the not-deformed tubular element, said annular wall laying on a plane orthogonal to the axis of the tubular element; c. placing the tubular element in a fixture located in a press, said fixture comprising a base plate and an upper plate, said base plate presenting a matrix die consisting of a cylindrical hollow featuring an outer diameter equal to the inner diameter of the outer end of the tubular element, whereas a special punch projects from said upper plate, said special punch being coaxial to said matrix die and consisting of an inner cylinder and an outer sleeve coaxial to each other and separated by a space, said cylinder featuring an outer diameter equal to the inner diameter of the inner end of the tubular element, and said sleeve featuring an inner diameter equal to the diameter of the tubular element in its not-deformed section; the tubular element being placed in the press coaxially to said matrix die and to said punch, so that the annular wall rests on the upper edge of said matrix die, whereas the outer end of the tubular element externally adheres to the matrix die; d. operating the press and starting the downward movement of the special punch, in this operating step the tubular element progressively inserting into the space determined inside said punch; e. causing the plastic deformation of the wall of the tubular element thanks to the stress exerted by the lower edge of the sleeve onto the wall of the tubular element, which is thus forced to assume the shape of the lateral surface of the sleeve; this way the outer end of the tubular element remaining adherent to the outer wall of the matrix die; f. lifting the punch, after the latter reaches its end-of-stroke position; g. taking the tubular element out of the press; h. cutting the outer end of the tubular element. a. reducing the diameter of an extreme section of said tubular element to form the inner end of the exhaust terminal; b. increasing the diameter of the opposite extreme section of said tubular element to form the outer end of the exhaust terminal, so as to determine an annulus which forms an annular wall joining said outer end to the not-deformed tubular element, said annular wall laying on a plane orthogonal to the axis of the tubular element; c. placing the tubular element in a fixture located in a press, said fixture comprising a base plate and an upper plate, said base plate presenting a matrix die consisting of a cylindrical hollow featuring an outer diameter equal to the inner diameter of the outer end of the tubular element, whereas a special punch projects from said upper plate, said special punch being coaxial to said matrix die and consisting of an inner cylinder and an outer sleeve coaxial to each other and separated by a space, said cylinder featuring an outer diameter equal to the inner diameter of the inner end of the tubular element, and said sleeve featuring an inner diameter equal to the diameter of the tubular element in its not-deformed section; the tubular element being placed in the press coaxially to said matrix die and to said punch, so that the annular wall rests on the upper edge of said matrix die, whereas the outer end of the tubular element externally adheres to the matrix die; d. operating the press and starting the downward movement of the special punch, in this operating step the tubular element progressively inserting into the space determined inside said punch; e. causing the plastic deformation of the wall of the tubular element thanks to the stress exerted by the lower edge of the sleeve onto the wall of the tubular element, which is thus forced to assume the shape of the lateral surface of the sleeve; this way the outer end of the tubular element remaining adherent to the outer wall of the matrix die; f. lifting the punch, after the latter reaches its end-of-stroke position; g. taking the tubular element out of the press; h. cutting the outer end of the tubular element.