We propose a new self-stabilizing 1-maximal matching algorithm that works under the distributed unfair daemon for arbitrarily shaped networks without cycle whose length is a multiple of three. The 1-maximal matching is a 2/3-approximation of a maximum matching, a significant improvement over the 1/2-approximation that is guaranteed by a maximal matching. Our algorithm is as efficient (its stabilization time is O(e) moves, where e denotes the number of edges in the network) as the best known algorithm operating under the weaker central daemon. It significantly outperforms the only known algorithm for the distributed daemon (with O(e) moves vs. O(2^n*δn) moves, where δ denotes the maximum degree of the network, and n its number of nodes), while retaining its silence property (after stabilization, its output remains fixed).