A spiral exchanger has a winding axis and comprises an outer sheet and an inner sheet secured to one another in a fastening plane before winding and delimiting a space for a fluid between them. The outer sheet and the inner sheet are wound on themselves and each comprises a plurality of flexible areas and a plurality of rigid areas, the flexible areas being more flexible than the rigid areas during folding. The flexible areas and the rigid areas are extended along the winding axis, and at least one flexible area of the outer sheet and at least one flexible area of the inner sheet that delimit the space between them form a pair of flexible areas that are aligned in a same radial direction.