To study the Gamow-Teller (GT) transitions to the pf-shell nucleus 64Cu, the 64Ni(3He,t)64Cu experiment was performed at the Research Center for Nuclear Physics Ring Cyclotron, Osaka, using a 3He beam of 140 MeV/nucleon. The outgoing tritons were momentum analyzed by the Grand Raiden spectrometer at 0°. A high energy resolution of 32 keV (fwhm) allowed the separation of individual levels in the excitation-energy region from 0 to 3.5 MeV. In addition to the ground state (gs), known to be a 1+ GT state, many low-lying states showed L=0 nature, suggesting that they are candidates for GT states. Because the GT strength B(GT) for the gs transition is known from the beta-decay measurement, the strengths for the excited states could be determined using the proportionality between the B(GT) and the reaction cross section extrapolated to q=0 momentum transfer. At higher excitation energies, the level density becomes high and the so-called GT giant resonance dominates the spectrum. The lower and the upper limits of the strength contained in this energy region were estimated. Our results show that less than 55% of the strength predicted by the Ikeda sum rule is located in the excitation-energy region from 0 to 17 MeV.