A method of laying an undersea line at the bottom of the sea from the surface to connect the bottom end thereof to a coupling element at the sea bottom by positioning the undersea line suspended from the surface in a substantially vertical position and including a curving device extending between two points and having at least one cable and tensioning means for tensioning the cable suitable for reducing the length of the cable extending between the two fastening or guide points from a maximum length L0 to a given shorter limit value L1; moving the bottom end of the undersea line in such a manner as to move the bottom end of the undersea line away from its initial position through a distance D, if necessary, moving the undersea line closer to the sea bottom while simultaneously lowering the line additionally from the surface, and then finalizing the positioning and connecting of the bottom end of the line to the sea bottom.