This paper presents a nonlinear finite-time stable attitude estimation scheme for a rigid body with unknown dynamics. Attitude is estimated from a minimum of two linearly independent known vectors measured in the body-fixed frame, and the angular velocity vector is assumed to have a constant bias in addition to measurement errors. Estimated attitude evolves directly on the special Euclidean group SO(3), avoiding any ambiguities. The constant bias in angular velocity measurements is also estimated. The estimation scheme is proven to be almost globally finite time stable in the absence of measurement errors using a Lyapunov analysis. For digital implementation, the estimation scheme is discretized as a geometric integrator. Numerical simulations demonstrate the robustness and convergence capabilities of the estimation scheme.