We prove that if A is a pseudo-differential operator with symbol in the Hörmander class S^m, m ≥ 0, considered as a closed linear operator on L²(Rⁿ), is invertible with A^-1 in B(L²(Rⁿ)), then A ^-1 is a pseudo-differential operator with symbol in s^-m. The proof entails only known results in C*-algebras, the known result that ellipticity is a consequence of Fredholmness for pseudo-differential operators with symbol in S^m and the well-known technique of transforming the class S^™ to the class S⁰.