An arrangement in the on-line finishing of a paper machine, which includes at least one finishing stage (12, 13) for the paper web produced, has cutting equipment (16) for forming the tail as well as tail threading equipment (15) for carrying the tail through the finishing stage (12, 13). At the end of the finishing stage (12, 13) there is a draw point (24), through which a carrier rope system (17) included in the tail threading equipment (15) is adapted to pass. The draw point (24) is formed between one cylinder (28) and a wire (29) arranged in contact with it for forming a single-contact draw point.