A process for piecing a broken yarn on an open-end spinning device in which the fiber feed to the fiber collection surface, and which has been interrupted during stoppage of the open-end spinning device. The fiber feed is switched back on and is immediately brought to full production speed. In this process the combed-out state of the fiber tuft at the point in time of when fiber feed is switched back on is ascertained. The point in time when yarn draw off is switched back on and the speed of it are adapted to when the feeding of fibers to the fiber collection surface becoming effective. The yarn is subjected to a multi-phase acceleration. The first phase of the draw-off acceleration is adapted to the incorporation of fibers into the back-fed yarn end. At least one additional phase of the draw-off acceleration serves to reach and/or to maintain the desired fiber mass. The control device to carry out the piecing process is connected to a device for the ascertainment of the combed-out state of the fiber tuft at the moment of piecing and the control device controls the yarn draw-off as a function of this ascertained combed-out state of the fiber tuft. At least two yarn accelerating devices are provided which can be selected by the control device.