A novel tissue-equivalent proportional counter (TEPC) based on a gas electron multiplier (GEM) for measuring H*(10) for neutrons was designed and constructed. The pulse height spectra (PHS) of two different neutron sources (a 252Cf source and a AmBe source) were measured using the new TEPC. The measurements were made with the TEPC filled with two different gases (10P gas and a propane-based tissue-equivalent gas) at various pressures. A computer simulation of the new TEPC, based on the Monte Carlo method, was performed to obtain the PHS for the two neutron sources. It is shown that the experimental results agree well with the simulation results for both 252Cf and AmBe neutron sources. Several outstanding problems are discussed and suggestions are made to make the GEM-based TEPC a practical neutron rem meter. The potential advantage of this novel neutron rem meter would be its low weight and compactness.