Stable ^{205}Tl ions have the lowest known energy threshold for capturing electron neutrinos (ν_{e}) of E_{ν_{e}}≥50.6 keV. The Lorandite Experiment (LOREX), proposed in the 1980s, aims at obtaining the longtime averaged solar neutrino flux by utilizing natural deposits of Tl-bearing lorandite ores. To determine the ν_{e} capture cross section, it is required to know the strength of the weak transition connecting the ground state of ^{205}Tl and the 2.3 keV first excited state in ^{205}Pb. …