  In this paper we analyze the stability of the traveling wave solution for an
ignition-temperature, first-order reaction model of thermo-diffusive
combustion, in the case of high Lewis numbers (${\rm Le} >1$). The system of
two parabolic PDEs is characterized by a free interface at which ignition
temperature $\Theta_i$ is reached. We turn the model to a fully nonlinear
problem in a fixed domain. When the Lewis number is large, we define a
bifurcation parameter $m=\Theta_i/(1-\Theta_i)$ and a perturbation parameter
$\varepsilon= 1/{\rm Le}$. The main result is the existence of a critical value
$m^c(\varepsilon)$ close to $m^c=6$ at which Hopf bifurcation holds for
$\varepsilon$ small enough. Proofs combine spectral analysis and non-standard
application of Hurwitz Theorem with asymptotics as $\varepsilon\to 0$.
