We address the problem of attitude and rate gyro bias estimation based on inertial and range data. Range measurements are obtained from beacons installed in an inertial frame and acoustic sensors fixed in the moving body reference frame. A nonlinear observer is proposed and its stability and performance properties are studied using Lyapunov techniques. In the presence of constant bias on the rate gyro measurements, the attitude and bias estimates are shown to converge exponentially to the desired values inside an almost global region of attraction. For time-varying bias with bounded derivative, a local result of ultimate boundedness is provided. The performance of the proposed nonlinear observer is validated experimentally using a high-accuracy calibration table, which provides ground-truth signals for comparison with the resulting estimates.