  The ground-state-to-ground-state $\beta$-decay $Q$-value of
$^{135}\textrm{Cs}(7/2^+)\to\,^{135}\textrm{Ba}(3/2^+)$ was directly measured
for the first time utilizing the Phase-Imaging Ion-Cyclotron Resonance (PI-ICR)
technique at the JYFLTRAP Penning-trap setup. It is the first direct
determination of this $Q$-value and its value of 268.66(30)\,keV is a factor of
three more precise than the currently adopted $Q$-value in the Atomic Mass
Evaluation 2016. Moreover, the $Q$-value deduced from the $\beta$-decay
endpoint energy has been found to deviate from our result by approximately 6
standard deviations. The measurement confirms that the first-forbidden unique
$\beta^-$-decay transition
$^{135}\textrm{Cs}(7/2^+)\to\,^{135}\textrm{Ba}(11/2^-)$ is a candidate for
antineutrino-mass measurements with an ultra-low $Q$-value of $0.44(31)$ keV.
This $Q$-value is almost an order of magnitude smaller than in any presently
running or planned direct (anti)neutrino-mass experiment.
