A neutron counter designed for assay of radioactive materials has been adapted for beam experiments at TUNL. The cylindrical geometry and 60% maximum efficiency make it well suited for ($gamma,n$) cross-section measurements near the neutron emission threshold. A high precision characterization of the counter has been made using neutrons from several sources. Using a combination of measurements and simulations, the absolute detection efficiency of the neutron counter was determined to an accuracy of $pm$ 3% in the neutron energy range between 0.1 and 1 MeV. It is shown that this efficiency characterization is generally valid for a wide range of targets.