An undersea pipe of the riser type providing a connection between a floating support and the sea bottom. The riser has a rigid pipe of the catenary type extending from the floating support to a point of contact with the sea bottom. The catenary riser comprises a lower pipe portion terminating at the point of contact. The apparent weight per unit length in water of the lower pipe portion being at least 25% less than that of the remainder of the pipe constituting the catenary riser.