The interpretation of observations of cooling neutron star crusts in quasipersistent x-ray transients is affected by predictions of the strength of neutrino cooling via crust Urca processes. The strength of crust Urca neutrino cooling depends sensitively on the electron-capture and β-decay ground-state-to-ground-state transition strengths of neutron-rich rare isotopes. Nuclei with a mass number of A=61 are predicted to be among the most abundant in accreted crusts, and the last remaining experimentally undetermined ground-state-to-ground-state transition strength was the β decay of ^{61}V. This Letter reports the first experimental determination of this transition strength, a ground-state branching of 8.1_{-3.1}^{+4.0}%, corresponding to a log ft value of 5.5_{-0.2}^{+0.2}. This result was achieved through the measurement of the β-delayed γ rays using the total absorption spectrometer SuN and the measurement of the β-delayed neutron branch using the neutron long counter system NERO at the National Superconducting Cyclotron Laboratory at Michigan State University. This method helps to mitigate the impact of the pandemonium effect in extremely neutron-rich nuclei on experimental results. The result implies that A=61 nuclei do not provide the strongest cooling in accreted neutron star crusts as expected by some predictions, but that their cooling is still larger compared to most other mass numbers. Only nuclei with mass numbers 31, 33, and 55 are predicted to be cooling more strongly. However, the theoretical predictions for the transition strengths of these nuclei are not consistently accurate enough to draw conclusions on crust cooling. With the experimental approach developed in this work, all relevant transitions are within reach to be studied in the future.