We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an $α$-stable law with $1<α<2$. We prove that the derivative martingale $D_n$ converges to a non-trivial limit $D_\infty$ under some regular conditions. We also study the additive martingale $W_n$, and prove $n^\frac{1}αW_n$ converges in probability to a constant multiple of $D_\infty$.