A process of the same class and a heat exchanger of the same class are known from German Offenlegungsschrift No. 33 38 157. In the known process, the tubes are introduced into a tube sheet in the form of a perforated sheet. Subsequently, the tube ends are welded to the tube sheet by means of an ultrasound head. The production of an ultrasonic horn (sonotrode) which is suitable for use as an ultrasound head is very costly. By means of the known process, each tube end must either be welded to the tube sheet individually, which is very inefficient, or one needs a specially-designed ultrasonic horn for each desired tube arrangement or number of tubes, respectively. Moreover, if the tube ends are welded individually, some previously-welded tube ends could become partially loose again, due to the thermal effect, while adjacent tube ends are being welded. The outer contour of the ultrasonic horns may be selected to be conical. If such an ultrasonic horn is used for welding the tube ends to the tube sheet, each tube end on the tube sheet of the finished heat exchanger will be conically-enlarged toward its end. This enlargement facilitates the admission and discharge of the medium flowing in the heat exchanger. Through these enlargements one achieves a reduction in the pressure drop during admission of the medium into the tubes, which is not necessarily desirable, for instance if the heat exchanger manufactured by this process is to be used as a vaporizer. The heat exchanger manufactured according to the known process exhibits an annular bulge at each tube exit on the front surface of the tube sheet. In addition, a distinct boundary between the tube material and the tube sheet material is visible on the front surface of the tube sheet.