The interpretation of how superconductivity disappears in cuprates at large hole doping has been controversial. To address this issue, we present an experimental study of single-crystal and thin film samples of La2−xSrxCuO4 (LSCO) with x≥0.25. In particular, measurements of bulk susceptibility on LSCO crystals with x=0.25 indicate an onset of diamagnetism at Tc1=38.5 K, with a sharp transition to a phase with full bulk shielding at Tc2=18 K, independent of field direction. Strikingly, the in-plane resistivity only goes to zero at Tc2. Inelastic neutron scattering on x=0.25 crystals confirms the presence of low-energy incommensurate magnetic excitations with reduced strength compared to lower doping levels. The ratio of the spin gap to Tc2 is anomalously large. Our results are consistent with a theoretical prediction for strongly overdoped cuprates by Spivak, Oreto, and Kivelson, in which superconductivity initially develops within disconnected self-organized grains characterized by a reduced hole concentration, with bulk superconductivity occurring only after superconductivity is induced by proximity effect in the surrounding medium of higher hole concentration. Beyond the superconducting-to-metal transition, local differential conductance measurements on an LSCO thin film suggest that regions with pairing correlations survive, but are too dilute to support superconducting order. Future experiments will be needed to test the degree to which these results apply to overdoped cuprates in general.