It has been reasoned that the structures of strongly cellular flames in very lean mixtures approach an array of flame balls, each burning as if it were isolated, thereby indicating a connection between the critical conditions required for existence of steady flame balls and those necessary for occurrence of self-sustained premixed combustion. This is the starting assumption of the present study, in which structures of near-limit steady sphericosymmetrical flame balls are investigated with the objective of providing analytic expressions for critical combustion conditions in ultra-lean hydrogen-oxygen mixtures diluted with N-2 and water vapor. If attention were restricted to planar premixed flames, then the lean-limit mole fraction of H-2 would be found to be roughly ten percent, more than twice the observed flammability limits, thereby emphasizing the relevance of the flame-ball phenomena. Numerical integrations using detailed models for chemistry and radiation show that a one-step chemical-kinetic reduced mechanism based on steady-state assumptions for all chemical intermediates, together with a simple, optically thin approximation for water-vapor radiation, can be used to compute near-limit fuel-lean flame balls with excellent accuracy. The previously developed one-step reaction rate includes a crossover temperature that determines in the first approximation a chemical-kinetic lean limit below which combustion cannot occur, with critical conditions achieved when the diffusion-controlled radiation-free peak temperature, computed with account taken of hydrogen Soret diffusion, is equal to the crossover temperature. First-order corrections are found by activation-energy asymptotics in a solution that involves a near-field radiation-free zone surrounding a spherical flame sheet, together with a far-field radiation-conduction balance for the temperature profile. Different scalings are found depending on whether or not the surrounding atmosphere contains water vapor, leading to different analytic expressions for the critical conditions for flame-ball existence, which give results in very good agreement with those obtained by detailed numerical computations. The one-step chemistry employed in the present work, which involves a non-Arrhenius rate having a cutoff at the crossover temperature, applies with excellent accuracy to the description of lean premixed hydrogen-air combustion, i.e, for phi less than or similar to 0.5 at atmospheric pressure, and could be used for instance in the numerical simulation of the propagation of curved or cellular flames in ultra-lean reactive atmospheres, of interest for safety analyses related to the storage, transport, and handling of hydrogen. Copyright (C) 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.