The doping of charge carriers into the CuO 2 planes of copper oxide Mott insulators causes a gradual destruction of antiferromagnetism and the emergence of high-temperature superconductivity. Optimal superconductivity is achieved at a doping concentration p beyond which further increases in doping cause a weakening and eventual disappearance of superconductivity. A potential explanation for this demise is that ferromagnetic fluctuations compete with superconductivity in the overdoped regime. In this case, a ferromagnetic phase at very low temperatures is predicted to exist beyond the doping concentration at which superconductivity disappears. Here 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 at low temperature in the heavily overdoped nonsuperconducting region. However, the magnetism does not exist in a commensurate long-range ordered state. Instead it appears as a dilute concentration of static magnetic moments. This finding places severe restrictions on the form of ferromagnetism that may exist in the overdoped regime. Although an extrinsic impurity cannot be absolutely ruled out as the source of the magnetism that does occur, the results presented here lend support to electronic band calculations that predict the occurrence of weak localized ferromagnetism at high doping.A ttempts within the framework of standard theories have failed to explain how high-temperature superconductivity emerges from charge carrier doping of an antiferromagnetic (AF) Mott insulator (1). The conventional Bardeen-CooperSchrieffer (BCS) theory of low-temperature superconductors (2) assumes that above the critical transition temperature (T c ) the electrons form a Landau Fermi liquid, and that superconductivity arises from pair condensation of the associated low-energy excitations (quasiparticles). What is clear in the case of copper oxides is that over the initial doping range where superconductivity first appears, ordinary Landau Fermi liquid theory does not apply. This realization has prompted theories in which the properties of the ordinary Fermi liquid are hidden (3), or alternatively only some properties of the Fermi liquid persist (4).A BCS-type theory may be applicable in the heavily overdoped region, where an ordinary Fermi liquid is observed (5-7). However, the recent proposal by Kopp et al. (8) that ferromagnetic (FM) fluctuations compete with d-wave superconductivity is a challenge to the notion that overdoped copper oxides strictly conform to such conventional wisdom. The primary motivation for their hypothesis is a strong upturn in the magnetic susceptibility immediately above T c for doping levels greater than p ∼ 0.19 (9-14). A tendency toward FM order for high charge doping (Fig. 1) is supported by electronic band calculations for super cells of La 2−x Ba x CuO 4 (15). However, these calculations favor the appearance of weak ferromagnetism about concentrated regions of ...