This article presents an adaptive antisaturation prescribed-time control (AAPTC) scheme for the stable payload retrieval of a two-body tethered space system following the desired libration state. The tethered payload is retrieved at the analytical speed function corresponding to the desired stable libration state. At the same time, the thrust at the payload is applied with the proposed AAPTC scheme to closely track the desired state to eliminate any deviation from the desired libration state due to external disturbances. The influence of the thrust saturation is minimized by a neural dynamics-based adaptive antisaturation compensator (ASC) that quickly drives the control system away from the saturated region. The Lyapunov stability of the proposed control scheme is proved analytically. Numerical simulations validate the effectiveness of the proposed prescribed-time strategy and the adaptive ASC.