The matching problem asks for a large set of disjoint edges in a graph. It is a problem that has received considerable attention in both the sequential and self-stabilizing literature. Previous work has resulted in self-stabilizing algorithms for computing a maximal ($\frac{1}{2}$-approximation) matching in a general graph, as well as computing a $\frac{2}{3}$-approximation on more specific graph types. In the following we present the first self-stabilizing algorithm for finding a $\frac{2}{3}$-approximation to the maximum matching problem in a general graph. We show that our new algorithm when run under the distributed adversarial daemon, stabilizes after at most $O(n^2)$ rounds. However, it might still use an exponential number of time steps.