In the production of filter-tipped cigarettes (25), an apparatus (filter attachment machine) is used, which consists of a main cylinder (10) and a plurality of assigned further cylinders. The elements of the filter-tipped cigarettes (25), specifically a tobacco rod (15) or part-rods (16, 17), filter pieces (21), cigarette units (23), and the filter-tipped cigarettes (25) produced therefrom are transported by the main cylinder (10), with the exception that part-rods (16, 17) displaced at a distance from one another and with a filter piece (21) inserted for the attachment of a wrapping paper (22) are transported and connected to one another outside the main cylinder (10), so that one-piece cigarette units (23) are returned to the main cylinder (10).