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[en] The newly built second experimental area EAR2 of the n-TOF spallation neutron source at CERN allows to perform (n, charged particles) experiments on short-lived highly radioactive targets. This paper describes a detection apparatus and the experimental procedure for the determination of the cross-section of the "7Be(n,α)α reaction, which represents one of the focal points toward the solution of the cosmological Lithium abundance problem, and whose only measurement, at thermal energy, dates back to 1963. The apparently unsurmountable experimental difficulties stemming from the huge "7Be γ-activity, along with the lack of a suitable neutron beam facility, had so far prevented further measurements. The detection system is subject to considerable radiation damage, but is capable of disentangling the rare reaction signals from the very high background. This newly developed setup could likely be useful also to study other challenging reactions requiring the detectors to be installed directly in the neutron beam.