A collaboration of radiation physicists from Tohoku University, CERN, and Stanford Linear Accelerator Center (SLAC) have completed a set of experiments to measure the neutron energy and time-of-flight spectra behind the lateral shield of the electron beam dump at the Final Focus Test Beam (FFTB) Facility at SLAC. The experiment received beam for ten days in 2001, and for another eight days in June 2002. Neutrons were produced by a 28.7-GeV electron beam hitting the aluminum beam dump of the FFTB, which is housed inside a thick steel and concrete shield. The measurements were performed using an NE213 organic liquid scintillator behind different thicknesses of the concrete shield of 274 cm, 335 cm, 396 cm, and 457 cm, respectively. The neutron energy spectra between 6 and 800 MeV were obtained by unfolding the measured pulse-height spectrum with the detector-response function. Neutrons below 20 MeV were also measured with a Bonner multisphere detector set. The corresponding experimental results have been simulated with the FLUKA Monte Carlo code. The collaboration has just submitted the first results from the experiment for publication.