The Gamow-Teller (GT) strength distribution has been investigated in a high-resolution (He-3,t) charge-exchange experiment on the double-beta (beta beta) decaying nucleus Zr-96. The experiment was carried out at the Osaka University Research Center for Nuclear Physics (RCNP) with a 420 MeV incident He-3 beam and the Grand Raiden magnetic spectrometer. A final-state energy resolution of 32 keV was achieved. A large fraction of the low-energy GT(-) strength is identified in a single state at 694 keV. A similar observation has been made in a Mo-96(d,He-2)Nb-96 experiment at 120 keV resolution for the GT(+) strength. It is found that this single state makes up nearly 100% of the nuclear matrix element for the Zr-96 two-neutrino double-beta (2 nu beta beta) decay.