This article addresses the distributed finite-time attitude tracking control problem for a group of uncertain rigid spacecraft in the presence of unavailable angular velocity under the directed topology condition. First, a finite-time adaptive neural network observer is proposed for each follower to estimate its own unavailable angular velocity. Unlike existing velocity-observer-based design methods, the proposed one does not need the exact knowledge of the system model, and works well for the systems with both vanishing and nonvanishing uncertainties. Further, another finite-time observer is provided to obtain the precise angular velocity information of the dynamic leader in a distributed manner. Based on these two observers and adding a power integrator technique, a continuous distributed finite-time control scheme with only attitude measurements is finally established. A rigorous theoretical proof shows that the entire finite-time stability of the combined observer-controller closed-loop system is ensured. Simulation results illustrate the benefits and effectiveness of the developed control scheme.