  We consider the following generalized derivative nonlinear Schr\"odinger
equation \begin{equation*}
i\partial_tu+\partial^2_xu+i|u|^{2\sigma}\partial_xu=0,\ (t,x)\in\mathbb
R\times\mathbb R \end{equation*} when $\sigma\in(0,1)$. The equation has a
two-parameter family of solitary waves
$$u_{\omega,c}(t,x)=\Phi_{\omega,c}(x)e^{i\omega t+\frac{ic}2x-\frac
i{2\sigma+2}\int_0^x\Phi_{\omega,c}(y)^{2\sigma}dy},$$ with $(\omega,c)$
satisfying $\omega>c^2/4$, or $\omega=c^2/4$ and $c>0$. The stability theory in
the frequency region $\omega>c^2/4$ was studied previously. In this paper, we
prove the stability of the solitary wave solutions in the endpoint case
$\omega=c^2/4$ and $c>0$.
