The energy-dependent cross section of the Be7(n,alpha)He4 reaction, of interest for the so-called cosmological lithium problem in big bang nucleosynthesis, has been measured for the first time from 10 meV to 10 keV neutron energy. The challenges posed by the short half-life of Be7 and by the low reaction cross section have been overcome at n_TOF thanks to an unprecedented combination of the extremely high luminosity and good resolution of the neutron beam in the new experimental area (EAR2) of the n_TOF facility at CERN, the availability of a sufficient amount of chemically pure Be7, and a specifically designed experimental setup. Coincidences between the two alpha particles have been recorded in two Si-Be7-Si arrays placed directly in the neutron beam. The present results are consistent, at thermal neutron energy, with the only previous measurement performed in the 1960s at a nuclear reactor. The energy dependence reported here clearly indicates the inadequacy of the cross section estimates currently used in BBN calculations. Although new measurements at higher neutron energy may still be needed, the n_TOF results hint at a minor role of this reaction in BBN, leaving the long-standing cosmological lithium problem unsolved.