An apparatus for continuously unwinding a plurality of rolled-up tapes made of fusible material wherein the tapes are successively unwound, one at a time, from at least two rewinders for supporting the tape thereon, with the tail end of the preceding tape joined to the leading end of the following tape. The apparatus comprises a guide roller assembly for transporting the preceding tape via a joining station, a tape end sensor for detecting the tail end of the preceding tape, and a joining device for holding the following tape at its leading end as folded over and opposed to the preceding tape, thermally bonding the tail end of the preceding tape to the folded-over leading end of the following tape in response to a detection signal from the tape end sensor, and cutting off a portion of the resulting joint with only a required length thereof left uncut.