We present the first polynomial self-stabilizing algorithm for finding a $\frac23$-approximation of a maximum matching in a general graph. The previous best known algorithm has been presented by Manne \emph{et al.} \cite{ManneMPT11} and has a sub-exponential time complexity under the distributed adversarial daemon \cite{Coor}. Our new algorithm is an adaptation of the Manne \emph{et al.} algorithm and works under the same daemon, but with a time complexity in $O(n^3)$ moves. Moreover, our algorithm only needs one more boolean variable than the previous one, thus as in the Manne \emph{et al.} algorithm, it only requires a constant amount of memory space (three identifiers and $two$ booleans per node).