We report on the measurement of the $^{7}$Be($n, p$)$^{7}$Li cross section from thermal to approximately 325 keV neutron energy, performed in the high-flux experimental area (EAR2) of the n_TOF facility at CERN. This reaction plays a key role in the lithium yield of the Big Bang Nucleosynthesis (BBN) for standard cosmology. The only two previous time-of-flight measurements performed on this reaction did not cover the energy window of interest for BBN, and showed a large discrepancy between each other. The measurement was performed with a Si-telescope, and a high-purity sample produced by implantation of a $^{7}$Be ion beam at the ISOLDE facility at CERN. While a significantly higher cross section is found at low-energy, relative to current evaluations, in the region of BBN interest the present results are consistent with the values inferred from the time-reversal $^{7}$Li($p, n$)$^{7}$Be reaction, thus yielding only a relatively minor improvement on the so-called Cosmological Lithium Problem (CLiP). The relevance of these results on the near-threshold neutron production in the p+$^{7}$Li reaction is also discussed.
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