A form-rolling die structure with die members which are pressed against a material to be threaded and which have rigid surfaces that are relatively displaced. The die members are equipped with a plurality of independent concave portions which form a substantially parallelogram shape when viewed from the normal of a virtual surface obtained by connecting the outermost portions of each of the rigid surfaces, and which are recessed from the virtual surface. Two of the corners at the sites corresponding to the four corners of the substantially parallelogram-shape of the concave portions are rounded. The periphery of each concave portion in the cross-section along the direction of the normal is rounded along the periphery of the substantially parallelogram shape. The concave portions form a substantially truncated-pyramid shape.