A state feedback H-infinity-suboptimal control problem on TSO(3) for a rigid spacecraft Kith three control torques and disturbances is addressed, The Hamilton-Jacobi inequality associated with a corresponding state feedback H-infinity-suboptimal control problem on TS3 is solved globally. The H-infinity-suboptimal state feedback on TS3 involves the Euler parameters and solves the original problem on TSO(3) when the initial values of the Euler parameters are appropriately chosen.