This study focuses on stabilizing the libration dynamics of an electrodynamic tether system (EDTS) using generalized torques induced by the Lorentz force. In contrast to existing numerical optimization methods, a novel analytical feedback control law is developed to stabilize the in-plane and out-of-plane motions of a tether by modulating the electric current only. The saturation constraint on the current is accounted for by adding an auxiliary dynamic system to the EDTS. To enhance the robustness of the proposed controller, multiple perturbations of the orbital dynamics, modeling uncertainties, and external disturbances are approximated using a neural network in which the weighting matrix and approximation error are estimated simultaneously, such that these perturbations are well compensated for during the control design of the EDTS. Furthermore, a dynamically scaled generalized inverse is utilized to address the singular matrix in the control law. The closed-loop system is proven to be ultimately bounded based on Lyapunov stability theory. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed analytical control law.