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.
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