  { The ground-state to ground-state electron-capture $Q$ value of $^{159}$Dy
($3/2^-$) has been measured directly utilizing the double Penning trap mass
spectrometer JYFLTRAP. A value of 364.73(19)~keV was obtained from a
measurement of the cyclotron frequency ratio of the decay parent $^{159}$Dy and
the decay daughter $^{159}$Tb ions using the novel phase-imaging ion-cyclotron
resonance technique. The $Q$ values for allowed Gamow-Teller transition to
$5/2^-$ and the third-forbidden unique transition to $11/2^+$ state with
excitation energies of 363.5449(14)~keV and 362.050(40)~keV in $^{159}$Tb were
determined to be 1.18(19) keV and 2.68(19) keV, respectively. The
high-precision $Q$ value of transition $3/2^-\to 5/2^-$ from this work,
revealing itself as the lowest electron-capture $Q$ value, is utilized to
unambiguously characterise all the possible lines that are present in its
electron capture spectrum. { We performed atomic many-body calculations for
both transitions to determine electron-capture probabilities from various
atomic orbitals, and found an order of magnitude enhancement in the event rates
near the end-point of energy spectrum in the transition to the $5/2^-$ nuclear
excited state, which can become very interesting once the experimental
challenges of identifying decays into excited states are overcome. The
transition to the $11/2^+$ state is strongly suppressed and found unsuitable
for measuring the neutrino mass. These results show that the electron capture
in the $^{159}$Dy atom, going to the $5/2^-$ state of the $^{159}$Tb nucleus,
%\textcolor{red} {is a new candidate which may open the way to determine the
electron-neutrino mass in the sub-eV region by studying EC. Further
experimental feasibility studies, including coincidence measurements with
realistic detectors, will be of great interest.} }
