  A precise determination of the atomic mass of $^{75}$As has been performed
utilizing the double Penning trap mass spectrometer, JYFLTRAP. The mass excess
is measured to be -73035.519(42) keV/c$^2$, which is a factor of 21 more
precise and 1.3(9) keV/c$^2$ lower than the adopted value in the newest Atomic
Mass Evaluation (AME2020). This value has been used to determine the
ground-state-to-ground-state electron-capture decay $Q$ value of $^{75}$Se and
$\beta^-$ decay $Q$ value of $^{75}$Ge, which are derived to be 866.041(81) keV
and 1178.561(65) keV, respectively. Using the nuclear energy-level data of
860.00(40) keV, 865.40(50) keV (final states of electron capture) and
1172.00(60) keV (final state of $\beta^-$ decay) for the excited states of
$^{75}$As$^*$, we have determined the ground-state-to-excited-state $Q$ values
for two transitions of $^{75}$Se $\rightarrow$ $^{75}$As$^*$ and one transition
of $^{75}$Ge $\rightarrow$ $^{75}$As$^*$. The ground-state-to-excited-state $Q$
values are determined to be 6.04(41) keV, 0.64(51) keV and 6.56(60) keV,
respectively, thus confirming that the three low $Q$-value transitions are all
energetically valid and one of them is a possible candidate channel for
antineutrino mass determination. Furthermore, the ground-state-to-excited-state
$Q$ value of transition $^{75}$Se $\rightarrow$ $^{75}$As$^*$ (865.40(50) keV)
is revealed to be ultra-low (< 1 keV) and the first-ever confirmed EC
transition possessing an ultra-low $Q$ value from direct measurements.
