Unwinding machine comprising two reelstand towers (1) covered by an overhead bridge (2) supporting a web accumulator (3), and two splicing cases (4) adapted to splice webs at zero speed each being linked to one of the reelstand towers (1) and rolling along rails fastened under the bridge (2). Each reelstand tower (1) is constituted by a frame (5) connected to the bridge (2) and a mobile frame (6) movable on the frame (5) perpendicularly to the bridge length (2), this frame (6) bearing also a device (7) to vertically move the cantilever spindle (8) to be inserted into the new reel core (9) to be unwound and following the transverse movements of the corresponding splicing case (4). The mobile frame (6) either follows rails on the frame (5), or rolls on this frame (5), the transverse movement being power controlled.