This paper proposes a selectively anti-disturbance control design for low-frequency flexible spacecraft while meeting strict convergence time constraints. Conditional disturbance negation (CDN) is investigated to make maximum use of disturbances by compensating harmful ones selectively while profiting from beneficial ones, which suppresses chattering in variables. Exact-time control (ETC) achieves exact and independent appointment of the convergence time by introducing a regulation function in the predefined-time control with neural networks (NN) removing singularity. ETC combined with CDN guarantees time accuracy and small chattering simultaneously. The globally exact-time stability analysis is conducted rigorously, and the boundedness of all variables can be achieved. Comparative simulations demonstrate the superiority and feasibility of the proposed controller.