We present an algorithm for the computation of the Kronecker structure of a symmetric pencil A-λM. This method, which preserves the property of symmetry, is somewhat different from Van Dooren's algorithm for the determination of the Kronecker structure of an arbitrary pencil. We show how to use this method to determine the structure of the infinite elementary divisors of A-λM and its Kronecker blocks, which may occur for the case of a singular pencil. The cost of computations of our algorithm is cheaper than Van Dooren's algorithm, and the symmetry of the reduced pencils can be preserved. The algorithm is fairly stable, though we use some nonunitary transformation matrices, but the norms of these matrices are bounded under a tolerance. The present procedure can also be used to separate from a symmetric pencil a smaller symmetric regular pencil, which contains only the finite eigenvalues of the original one; so this method can be used as a "preprocessing" for a QZ or HR algorithm in order to get rid of singularities, which can be troublesome for those algorithms.