Emergence of a Novel Frozen Magnetic State in Heavily Over-Doped La$_{2-x}$Sr$_{x}$CuO$_{4}$*

It has been proposed that the termination of superconductivity in heavily overdoped copper oxides is caused by the emergence of competing ferromagnetism. We report on a direct examination of this scenario in overdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$ using the technique of muon spin relaxation. We detect the onset of static magnetic moments of electronic origin beyond the superconducting ``dome'', where antiferromagnetic correlations are absent. The magnetism does not exist in a commensurate long-range ordered state, but instead appears as a dilute concentration of static magnetic moments. The results are qualitatively consistent with electronic band calculations that predict weak ferromagnetic order about Sr-rich clusters.