A possible explanation of the so-called "Cosmological Lithium Problem", an important unsolved problem in Nuclear Astrophysics, involves large systematic uncertainties in the cross-sections of reactions leading to the destruction of 7 Be during the Big-Bang Nucleosynthesis (BBN). Among these reactions, the 7 Be(n,) is the most uncertain. So far, only a single measurement with thermal neutrons has been performed. Therefore, BBN calculations had to rely on rather uncertain theoretical extrapolations. The short half-life of 7 Be (53.29 d) and the low cross section have prevented, up to now, to obtain experimental data at keV neutron energies typical for BBN studies. We have measured for the first time at n_TOF the 7 Be(n,) reaction in a wide neutron energy range, from thermal up to 10 keV. This measurement has been performed, at the new beam line (EAR2) of the Neutron-Time-Of-Flight facility n_TOF at CERN. The two -particles, emitted back-to-back in the reaction, have been detected by mean of sandwiches of silicon detectors and, by exploiting the coincidence technique, we were able to suppress the large and n-induced background. The 7 Be isotope production and purification has been performed by PSI-Zurich Switzerland.