We propose a simple “geometry-free” nonlinear observer using as measurements the biased angular velocity in body axes (given by a triaxial rate gyro), together with the expression in body axes of a single vector constant in inertial axes. Provided there is sufficient angular excitation, which ensures observability, the observer can estimate the measured direction -i.e., two of the three Euler angles of the orientation matrix- as well as the three angular velocity biases. The convergence is global and uniform, thanks to a strict (and nontrivial) Lyapunov function. Since all three biases are estimated, it is then possible to get a reasonable estimation of another direction, hence of the full attitude, by open-loop integration; indeed, as the gyro biases are taken into account, the integration drift is moderate since solely due to the gyro noise. The estimate so obtained may be sufficient in many practical control situations. Notice it is not possible to get a better result for observability reasons. The approach is illustrated by a detailed numerical simulation.