This is a mathematical study of the long-lived hydromagnetic structures produced in the tenuous solar corona by the turbulent, resistive relaxation of a magnetic field under the condition of extremely high electrical conductivity. The relaxation theory of Taylor, originally developed for a laboratory device, is extended to treat the open atmosphere where the relaxing field must interact with its surrounding fields. A boundary-value problem is posed for a two-dimensional model that idealizes the corona as the half Cartesian plane filled with a potential field (1) that is anchored to a rigid, perfectly conducting base and (2) that embeds a force-free magnetic field in the form of a flux-rope oriented horizontally and perpendicular to the Cartesian plane. The flux-rope has a free boundary, which is an unknown in the construction of a solution for this atmosphere. Pairs of magnetostatic solutions are constructed to represent the initial and final states of a flux-rope relaxation that conserve both the total magnetic helicity and total axial magnetic flux, using a numerical iterative method specially developed for this study. The collection of numerical solutions found provides an insight into the interplay among several hydromagnetic properties in the formation of long-lived coronal structures. In particular, the study shows (1) that the outward spread of reconnection between a relaxing flux-rope and its external field may be arrested at some outer magnetic flux surface within which a constant-α force-free field emerges as the minimum-energy state and (2) that this outward spread is complicated by an inward, partial collapse of the relaxing flux-rope produced by a loss of internal magnetic pressure.