Electron capture and beta decay play important roles in the evolution of presupernovae stars and their eventual core collapse. These rates are normally predicted through shell-model calculations. Experimentally determined strength distributions from charge-exchange reactions are needed to test modern shell-model calculations. We report on the measurement of the Gamow-Teller strength distribution in Co-58 from the Ni-58(t,He-3) reaction with a secondary triton beam of an intensity of similar to 10(6) pps at 115 MeV/nucleon and a resolution of similar to 250 keV. Previous measurements with the Ni-58(n,p) and the Ni-58(d,He-2) reactions were inconsistent with each other. Our results support the latter. We also compare the results to predictions of large-scale shell-model calculations using the KB3G and GXPF1 interactions and investigate the impact of differences between the various experiments and theories in terms of the weak rates in the stellar environment. Finally, the systematic uncertainties in the normalization of the strength distribution extracted from Ni-58(He-3, t) are described and turn out to be nonnegligible due to large interferences between the Delta L=0,Delta S=1 Gamow-Teller amplitude and the Delta L=2,Delta S=1 amplitude.