This paper describes the Plateau de Bure Neutron Monitor (PdBNM), an instrument providing continuous ground-level measurements of atmospheric secondary neutron flux resulting from the interaction of primary cosmic rays with the Earth's atmosphere. The detector is installed on the Plateau de Bure (Devoluy mountains, south of France, latitude North 44 degrees 38' 02 `', longitude East 5 degrees 54' 26 `', altitude 2555 m) as a part of the ASTEP Platform (Altitude Single-event effects Test European Platform), a permanent installation dedicated to the study of the impact of terrestrial natural radiation on microelectronics circuit reliability. The present paper reports the neutron monitor design, its operation since August 2008 and its complete numerical simulation using the Monte Carlo codes GEANT4 and MCNPX. We particularly detail the computation of the neutron monitor detection response function for neutrons, muons, protons and pions, the comparison between GEANT4 and MCNPX numerical results and the evaluation of the PdBNM counting rate as function of both the nature and flux of the incident atmospheric particles.