In this paper we have reconstructed an armillary sphere based on the Method of an Armillary Sphere Making described in the Volume 1 of The Collection of Writings on the Scientific Instruments-Uigijipsol (儀器輯說, two volumes) edited in the 1850's by Nam Byong-Chul (南秉哲, 1817-1863) who was a famous Korean states-man-scientist. Nam achieved convenience and accuracy in the measurements of stellar positions in the manner of selective setting the equatorial, ecliptic and horizontal poles by adding a pole axis exchange ring called Jaigeukkwon (載極圈) between the Three Arrangers of Time and Four Displacements. We made use of 3-dimensional graphic software for modelling Nam's armillary sphere which consisted of five layers-eight rings. Results of simulation showed that the pole axis exchange ring functioned properly in setting the equatorial, ecliptic and horizontal coordinates simply by exchange of positions of specified holes on the ring. We ascertained that the invention of Jaigeukkwon solved inherent problems in the conventional Chinese armillary sphere in computation of real ecliptic coordinates. It was revealed that Nam Byong-Chul made great contributions in the East Asian history of armillary sphere making.