We report the first detection of the second-forbidden, non-unique, $2^+rightarrow 0^+$, ground-state transition in the $beta$ decay of $^{20}$F. A low-energy, mass-separated $^{20}rm{F}^+$ beam produced at the IGISOL facility in Jyvaskyla, Finland, was implanted in a thin carbon foil and the $beta$ spectrum measured using a magnetic transporter and a plastic-scintillator detector. The $beta$-decay branching ratio inferred from the measurement is $b_{beta} = [ 0.41pm 0.08textrm{(stat)}pm 0.07textrm{(sys)}] times 10^{-5}$ corresponding to $log ft = 10.89(11)$, making this one of the strongest second-forbidden, non-unique $beta$ transitions ever measured. The experimental result is supported by shell-model calculations and has significant implications for the final evolution of stars that develop degenerate oxygen-neon cores. Using the new experimental data, we argue that the astrophysical electron-capture rate on $^{20}$Ne is now known to within better than 25% at the relevant temperatures and densities.
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