Solving the puzzle of the absolute mass scale of neutrinos is an outstanding issue of paramount importance. Current approaches that can directly pinpoint the (anti)neutrino mass in a precise and model-independent way are based on beta decay and electron capture experiments. Such experiments focus on decays that have the smallest decay energy -- $Q$-value -- to maximize the sensitivity to the neutrino mass. Here we report the ground-to-ground state electron-capture $Q$-value of $^{159}$Dy, measured directly for the first time using high-precision Penning trap mass spectrometry. The result, 364.73(19)~keV, reveals a decay channel to a state with spin-parity $5/2^-$ with the smallest $Q$-value of any known electron capture, 1.18(19)~keV. Investigation of the spectrum shape unveiled an order-of-magnitude enhancement in the event rate near the endpoint for $^{159}$Dy compared to $^{163}$Ho, which is so far the only nucleus used for direct neutrino mass determination. $^{159}$Dy is not only suitable but, by far, the best of any currently known ground-to-excited state decay candidate to pursue a neutrino mass measurement.
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