We investigate a new method to search for keV-scale sterile neutrinos that could account for Dark Matter. Neutrinos trapped in our galaxy could be captured on stable $^{163}$Dy if their mass is greater than 2.83 keV. Two experimental realizations are studied, an integral counting of $^{163}$Ho atoms in dysprosium-rich ores and a real-time measurement of the emerging electron spectrum in a dysprosium-based detector. The capture rates are compared to the solar neutrino and radioactive backgrounds. An integral counting experiment using several kilograms of $^{163}$Dy could reach a sensitivity for the sterile-to-active mixing angle $sin^2theta_{e4}$ of $10^{-5}$ significantly exceeding current laboratory limits. Mixing angles as low as $sin^2theta_{e4} sim 10^{-7}$ / $rm m_{^{163}rm Dy}rm{(ton)}$ could possibly be explored with a real-time experiment.
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