The structure of freely propagating, planar and adiabatic premixed flames is investigated for a simple two-step chain-branching model with finite activation energy within the diffusive-thermalpproximation. This article is focused on the flame structure near the flammability limit where the phenomenon of appreciable fuel leakage takes place. The asymptotic analysis was carried out with the help of the method of matched asymptotic expansions using a large parameter inversely proportional to the square of the flame propagation velocity S-L. Two terms of the corresponding expansions were obtained. The leading term provided the flammability limit condition in analytical form. The existence of multiple solutions for flames with fuel Lewis number larger than one was demonstrated analytically. The results of the asymptotic analysis were successfully compared with numerical computations. (c) 2012 The Combustion Institute. Published by Elsevier Inc. All rights reserved.