The cross section for the 4 He(α,n) 7 Be reaction was measured at low energies between E α = 38.50 and 39.64 MeV to solve the cosmological lithium problem. On the basis of the detailed balance principle, the cross section for the 7 Be(n,α) 4 He reaction was obtained at E c.m. = 0.20-0.81 MeV close to the Big Bang nucleosynthesis (BBN) energy window for the first time. The obtained cross sections are significantly smaller than the theoretical estimation used in the BBN calculations. The present results suggest the 7 Be(n,α) 4 He reaction rate is not large enough to solve the cosmological lithium problem.