Evaporation proton yields were measured for the fusion of the radioactive proton-rich nucleus Be7 onto a Ni58 target at six near-barrier energies. Total fusion cross sections were deduced by using calculated proton multiplicities. The resulting fusion excitation function shows a considerable enhancement with respect to calculations for a bare potential, even for energies above the Coulomb barrier. Inelastic couplings can account for the enhancement at the highest energy. Total fusion channels nearly saturate the total reaction cross section in the measured energy region. Comparison with previous results scaled appropriately for 7Be+(27Al,238U) shows good agreement.