The problem of the combustion of a methane–air mixture in a slot burner with adiabatic outer walls is solved. The problem is investigated numerically in a one-dimensional formulation and dimensional variables. The range of existence of stable high-temperature combustion of the methane–air mixture is determined as a function of gas flow rate and methane content in the mixture. The mechanism of failure of stable combustion is determined. The effect of heat transfer of the gas insert with an inner insert on combustion stability is shown.