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This paper presents a novel nonlinear adaptive neural control methodology for the challenging problem of deep-space spacecraft formation flying. By utilizing the framework of the circular restricted three-body problem with the Sun and Earth as the primary gravitational bodies, a nonlinear model is first developed, which describes the relative formation dynamics. This model is not confined to the vicinity of the Lagrangian libration points but rather constitutes the most general nonlinear formulation. Then, a relative position controller is designed, which consists of an approximate dynamic model inversion, linear compensation of the ideal feedback linearized model, and an adaptive neural network based element designed to compensate for the model inversion errors. The approach is illustrated by simulations, which confirm that the suggested methodology yields excellent tracking and disturbance rejection, thus permitting sub-millimeter formationkeeping precision.