We develop a new approach for determining the orientation of the axis (z) of finite-¿ magnetic flux ropes from single-spacecraft data. It consists of an optimization procedure based on two-dimensional magnetohydrostatic theory. From the requirement that the transverse pressure, Pt = p + Bz2/2¿0, be a function of the calculated magnetic potential, A, alone, an optimal z axis is found. Benchmark studies using analytical flux rope solutions show the feasibility of this axis-determination approach. Two magnetic cloud events, on 18 October 1995 and 9 January 1997, observed by the Wind spacecraft at 1 AU, are examined. The time series of plasma and magnetic field data, collected when the magnetic structures move past the spacecraft, are utilized as spatial initial values for numerical integration of the nonlinear, plane Grad-Shafranov equation. The recovered magnetic cloud structures show helical magnetic field configurations, representative of magnetic flux ropes, with invariance along their axes. Their cross sections consist of nested irregular loops of transverse field lines rather than the concentric circles of an axially symmetric model. In addition to their axis orientations, we obtain other quantitative features of the two magnetic clouds from their recovered cross sections, including their sizes (diameters ~ 0.2 AU), their impact parameters (|y0| ≲ 0.01 AU), their maximum axial field strengths (20 and 14 nT, respectively), and the twist of field lines (~ 1.30--2.68 turns AU-1) around the flux rope's axis. We find that errors in the orientation of the flux rope axis may result in significant variations in the main parameters, i.e., the impact parameter, as well as the size and shape of the cross section.