We have measured the overall detection efficiency to neutrons of a small prototype of the KLOE Pb-scintillating fiber calorimeter in the kinetic energy range 5-175 MeV using the neutron beam facility of The Svedberg Laboratory, TSL, Uppsala. The measurement of the neutron detection efficiency of a NE110 scintillator provided a reference calibration. At the lowest trigger threshold, the overall calorimeter efficiency ranges from 28% to 33%. This value largely exceeds the estimated ~8% expected if the response were proportional only to the scintillator equivalent thickness. A detailed simulation of the calorimeter and of TSL beamline has been performed with the FLUKA Monte Carlo code. The simulated response of the detector to neutrons is presented together with first data to Monte Carlo comparison. The results show an overall neutron efficiency of about 35%. The reasons for such an efficiency enhancement, in comparison with the typical scintillator-based neutron counters, are explained, opening the road to a novel neutron detector.