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 $\prod_{n=1}^{N} (1+βη_{n,S_n})$, where $(η_{n,x})_{n\ge 0, x\in \mathbb{Z}^d}$ is a field of i.i.d. random variables whose distribution satisfies $\mathbb{P}(η>z) \sim z^{-α}$ as $z\to\infty$, for some $α\in(0,2)$. We prove that if $α< \min(1+\frac{2}{d},2)$, when sending $N$ to infinity and rescaling the disorder intensity by taking $β=β_N \sim N^{-γ}$ with $γ=\frac{d}{2α}(1+\frac{2}{d}-α)$, the distribution of the trajectory under diffusive scaling converges in law towards a random limit, which is the continuum polymer with Lévy $α$-stable noise constructed in the companion paper arXiv:2007.06484.