A substrate sheet transporting device is provided. The device includes a first transfer cylinder defining a first axis, a circumferential direction and a transverse direction and a transfer belt (74) extending around the first transfer cylinder. The first transfer cylinder includes first openings for the passage of a gas and the transfer belt includes second openings for the passage of a gas. The transfer belt and the first transfer cylinder define an alignment position in which the second passage openings are aligned with the first passage openings. The transfer device includes a device for adjusting the position of the transfer belt relative to the first transfer cylinder by bringing the transfer belt toward the alignment position. The invention is applicable to cutting devices of rotary printing presses.