The ground state to ground state electron-capture Q value of 159Dy (3/2−) has been measured directly using 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 159Dy and the decay daughter 159Tb 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 159Tb were determined to be 1.18(19) keV and 2.68(19) keV, respectively. The high-precision Q value of transition 3/2−→5/2− from this work, revealing itself as the lowest electron-capture Q value, is used to unambiguously characterize 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 159Dy atom, going to the 5/2− state of the 159Tb nucleus, is a new candidate that may open the way to determine the electron-neutrino mass in the sub-eV region by studying electron-capture. Further experimental feasibility studies, including coincidence measurements with realistic detectors, will be of great interest.