A long-debated issue concerning the nucleosynthesis of neutron-rich elements in asymptotic giant branch (AGB) stars is the identification of the neutron source. We report intermediate-mass (4 to 8 solar masses) AGB stars in our Galaxy that are rubidium-rich as a result of overproduction of the long-lived radioactive isotope 87Rb, as predicted theoretically 40 years ago. This finding represents direct observational evidence that the 22Ne(α,n)25Mg reaction must be the dominant neutron source in these stars. These stars challenge our understanding of the late stages of the evolution of intermediate-mass stars and would have promoted a highly variable Rb/Sr environment in the early solar nebula.