The cuprate high-temperature superconductors (HTSC) have been the subject of intense study for more than 30 years with no consensus yet on the underlying mechanism of the superconductivity. Conventional wisdom dictates that the mysterious and extraordinary properties of the cuprates arise from doping a strongly correlated antiferromagnetic (AFM) insulator (1,2). The highly overdoped cuprates$-$those beyond the dome of superconductivity (SC)--are considered to be conventional Fermi liquid metals (3). Here, we report the emergence of itinerant ferromagnetic order (FM) below 4K for doping beyond the SC dome in electron-doped La$_{2-x} $Ce$_x$CuO$_4$ (LCCO). The existence of this FM order is evidenced by negative, anisotopic and hysteretic magnetoresistance, hysteretic magnetization, and the polar Kerr effect, all of which are standard signatures of itinerant FM in metals (4,5). This surprising new result suggests that the overdoped cuprates are also influenced by electron correlations and the physics is much richer than that of a conventional Fermi liquid metal.