A state observer is proposed for rigid body attitude motion with a given attitude dynamics model. This observer is designed on the state space of rigid body attitude motion, which is the tangent bundle of the Lie group of rigid body rotations in three dimensions, SO(3), and therefore avoids instability due to the unwinding phenomenon seen with unit quaternion-based attitude observers. In the absence of measurement noise and disturbance torques, the observer designed leads to almost global finite-time stable convergence of attitude motion state estimates to the actual states for a rigid body whose inertia is known. Almost global finite-time stability of this observer is shown using a Morse function as part of a Lyapunov analysis; this Morse function has been previously used for almost global asymptotic stabilization of rigid body attitude motion. Numerical simulation results confirm the analytically obtained stability properties of this attitude state observer. Numerical results also show that state estimate errors are bounded in the presence of bounded measurement noise and bounded disturbance torque.