This paper defines the stable topological cyclic homology $TC^S (R; M)$ of a simplicial ring $R$ with coefficients in an $R$-bimodule $M$. Moreover, it is proved that after profinite completion this construction equals the topological Hochschild homology $T(R; M)$ of $R$ with coefficient in $M$. This result should be compared to a recent result by B. Dundas and R. McCarthy which states that also the stable K-theory $K^S(R; M)$ equals topological Hochschild homology. The proof is based on a splitting of the topological Hochschild homology of the split extension of $R$ by the square zero ideal $M$ as a wedge of certain new functors in $R$ and $M$, one for each $n \geq 0$. It is shown that, in order to evaluate $TC^S(R; M)$, it suffices to consider the summands $n = 0, 1$. These are then identified as $T(R)$, resp. $S^1_+\wedge T(L; R)$.