  A high-precision measurement of the $^{131}$Cs$ \rightarrow ^{131}$Xe
ground-to-ground-state electron-capture $Q_{\textrm{EC}}$-value was performed
using the ISOLTRAP mass spectrometer at ISOLDE/CERN. The novel PI-ICR technique
allowed to reach a relative mass precision $\delta m/m$ of $1.4\cdot10^{-9}$. A
mass resolving power $m/\Delta m$ exceeding $1\cdot10^7$ was obtained in only
$1\,$s trapping time. Allowed electron-capture transitions with sub-keV or
lower decay energies are of high interest for the direct determination of the
$\nu_e$ mass. The new measurement improves the uncertainty on the
ground-to-ground-state $Q_{\textrm{EC}}$-value by a factor 25 precluding the
$^{131}$Cs$ \rightarrow ^{131}$Xe pair as a feasible candidate for the direct
determination of the $\nu_e$ mass.
