  We report the first direct determination of the ground-state to ground-state
electron-capture $Q$-value for the $^{72}$As to $^{72}$Ge decay by measuring
their atomic mass difference utilizing the double Penning trap mass
spectrometer, JYFLTRAP. The $Q$-value was measured to be 4343.596(75)~keV,
which is more than a 50-fold improvement in precision compared to the value in
the most recent Atomic Mass Evaluation 2020. Furthermore, the new $Q$-value was
found to be 12.4(40)~keV (3.1 $\sigma$) lower. With the significant reduction
of the uncertainty of the ground-state to ground-state $Q$-value value combined
with the level scheme of $^{72}$Ge from $\gamma$-ray spectroscopy, we confirm
that the five potential ultra-low $Q$-value ${\beta^{+}}$-decay or electron
capture transitions are energetically forbidden, thus precluding all the
transitions as possible candidates for the electron neutrino mass
determination. However, the discovery of small negative $Q$-values opens up the
possibility to use $^{72}$As for the study of virtual $\beta$-$\gamma$
transitions.
