A finite-time stable guidance law is proposed to deal with the three-dimensional terminal interception impact angle constraints problem where the accelerations of target are not available. A kind of non-linear observer is designed to estimate target's accelerations with the positions and velocities information given by missile. Further theoretical analysis is introduced to prove this observer's accuracy. Then the finite-time guidance law is deduced by sliding-mode control laws and finite-time stability theory using the observer estimated values, by which the convergence of line of sight (LOS) angles in three-dimensional space to expected impact angles in finite time is guaranteed. Through analyzing the effect of observer's errors on the guidance system, it is shown that the finite-time convergence can be achieved for the engineering practice by the proposed composite guidance law. Simulation results for uniformly/non-uniformly accelerated targets and the comparison with Proportion Navigation (PN) are provided to demonstrate the effectiveness of the proposed approach. The results suggest that difficulties of finite-time three-dimensional terminal interception problem with impact angle constraints can be solved by the proposed method combining the non-linear observer of target's accelerations with the finite-time sliding-mode control laws.