A process for manufacturing a line of variable stiffness over part of the length thereof and the associated element which allows the line to have the variable stiffness thereof is disclosed. The element is integrated during the manufacturing of the line or deposited thereafter around the line; the element consists of composite material underlayers in which the directions of the fibers have at least one component making a zero or a low-value angle with the axis of the pipe, and the layers end in a graded way. The process has application to the manufacturing of a riser base.