For spacecrafts with long operational life, the development of reserve ways for maintaining exact three-axis orientation, when there is no opportunity to use measurements of the full state vector, is a topical problem. In such cases, state observers are usually applied for asymptotic estimation (restoration) of unmeasured components of the state vector and subsequent use of the received estimations in the feedback path. In this article, we consider the problem of asymptotic estimation of the three angles of orientation, which are inaccessible to direct measurement, in the fine orientation mode in the orbital reference system by means of a state observer. Data from the gyroscopic angular velocity measuring device and from the sensor of a local vertical are used in the observer equations. Feedback coefficients for the observer, which provide attenuation of the estimation error, are obtained. Efficiency of the observer is validated by numerical simulation of the control system with the observer designed for a satellite with elastic elements. For the case of failure of the gyroscopic angular velocity measuring device, we also consider the problem of determination of angular velocity vector components on the basis of measurements of the angles of orientation obtained from high-precision star sensors. Here, the state observer is proposed to compute derivatives of the measured angles via polynomial approximation of measurements by using the method of least squares.