We propose a method for the stabilization of quantum computations (including quantum state storage). The method is based on the operation of projection into $\cal SYM$, the symmetric subspace of the full state space of R redundant copies of the computer. We describe an efficient algorithm and quantum network effecting $\cal SYM$--projection and discuss the stabilizing effect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed.