  In this paper, we study the so-called intermediate disorder regime for a
directed polymer in a random environment with heavy-tail. Consider a simple
symmetric random walk $(S_n)_{n\geq 0}$ on $\mathbb{Z}^d$, with $d\geq 1$, and
modify its law using Gibbs weights in the product form $\prod_{n=1}^{N}
(1+\beta\eta_{n,S_n})$, where $(\eta_{n,x})_{n\ge 0, x\in \mathbb{Z}^d}$ is a
field of i.i.d. random variables whose distribution satisfies
$\mathbb{P}(\eta>z) \sim z^{-\alpha}$ as $z\to\infty$, for some
$\alpha\in(0,2)$. We prove that if $\alpha< \min(1+\frac{2}{d},2)$, when
sending $N$ to infinity and rescaling the disorder intensity by taking
$\beta=\beta_N \sim N^{-\gamma}$ with $\gamma
=\frac{d}{2\alpha}(1+\frac{2}{d}-\alpha)$, the distribution of the trajectory
under diffusive scaling converges in law towards a random limit, which is the
continuum polymer with L\'evy $\alpha$-stable noise constructed in the
companion paper arXiv:2007.06484.
