This paper presents the design, analysis, and performance evaluation of a novel cascade observer for attitude estimation. A sensor-based observer, which resorts to rate gyro readings and a set of vector observations, estimates the rate gyro bias. Afterwards, a cascaded observer explicitly estimates the attitude in the form of a rotation matrix based on the rate gyro measurements, the vector observations, and the estimated gyro bias. The overall error dynamics are globally exponentially stable and the proposed system is computationally efficient. Finally, the resulting estimator is successfully evaluated, in simulation and experimentally, with ground truth data in both cases.