We report on a new cross-section measurement for the 3He(α,γ)7Be reaction at three medium energies of Ec.m. between 1 and 3 MeV. The interest stems from the significant role played by the reaction in calculating an accurate solar neutrino flux and the primordial 7Li abundance. The energy dependence of the astrophysical S34 factor observed in the present work, especially above 1 MeV, highlights the need to constrain theories in order to obtain a precise extrapolated value for S34(0). In this context, a comparison with the recent theoretical work in a fully microscopic fermionic molecular dynamics approach and a few other representative calculations emphasize the need for further experimental as well as theoretical work to resolve the existing conflicts.