The article studies the problem of three-axis reorientation of a rigid spacecraft. Three reaction wheels are employed to produce necessary torque in the axes of the spacecraft. External uncontrolled disturbances, that have no statistical description, are taken into consideration in the process of reorientation. Such a statement of the problem is characteristic for game theory. The controlling moments, applied to flywheels, are offered to be generated by means of a feedback inform of nonlinear functions of phase variables of a considered conflict-controlled system of differential equations, including dynamic Euler equations and kinematic equations in Rodrigues-Hamilton variables (in terms of the quaternion). As a result, the solution of the original nonlinear game problem is narrowed down to solving the elementary linear game problems. Since the exact solution of nonlinear game problems is very difficult, the proposed method seems to be effective. We stress the fact that the linearization used in n this case does not neglect nonlinear phenomena of the problem. Indeed, the converse passage from auxiliary control moments to initial ones, as well as the verification of constraints on control moments, requires the account of essentially nonlinear relations. The obtained control moments ensure the exact reorientation of the solid at a finite time. The reorientation being achieved by one spatial turn without causing additional constraints to the character of the resulting motion. The estimates of the accepted level of external disturbances, depending on the constraints on controlling moments and the initial location of the solid are found. These estimates are convenient at the first stage of a solution process, when the possibility is determined of using the proposed construction of control moments on order to ensure the required reorientation. If aforementioned estimates hold with a "reserve", then (at the second stage of the solution) a particular value of the guaranteed time of reorientation can be found by a special iteration algorithm.