In this research work, we report on the numerical predictions and analysis of stable, stationary and closed burner-stabilized reacting fronts under terrestrial-gravity conditions for ultra-lean hydrogen–methane–air premixed mixtures with a 40% hydrogen (H2) and 60% methane (CH4) fuel composition, specified on a molar basis. The transition from a cap-like to ball-like flame shape with decreasing inlet equivalence ratio is predicted in agreement with experimental observations. The predicted flames are compared to both flames that were studied in experiments and numerical solutions of perfectly-spherical flame balls in the absence of gravity and convection. The comparison includes flame size, lean limits, and when pertinent, standoff distances, all for two different reaction mechanisms. The absolute molar consumption rates of both H2 and CH4 for the limit flame attain maximum values that are significantly larger than those of the corresponding gravity-free flame ball. The fuel supply mechanism of the normal-gravity limit flame is similar to the fuel supply of flame balls in that it is driven by diffusion even away from the flame front. Heat conduction to the tube wall of the burner and convective heat loss are the dominant forms of heat loss. Furthermore, simulations with inclusion of multicomponent transport and Soret and Dufour effects show that the flame size increases for both flame balls and the burner-stabilized flames. For the latter, a slight modification in the stabilization position is found owing to the intensification of the consumption rates of both H2 and CH4 when these effects are accounted for. In summary, the present work considers a new configuration that allows the study of stable and stationary ball-like flames at ultra-lean and near-limit conditions, and advances the understanding of such flames via detailed numerical computations.