This paper is concerned with the Stein's method associated with a (possibly) asymmetric $α$-stable distribution $Z$, in dimension one. More precisely, its goal is twofold. In the first part, we exhibit a genuine bound for the Wasserstein distance between $Z$ and any integrable random variable $X$, in terms of an operator that reduces to the classical fractional Laplacian in the symmetric case. Then, in the second part we apply the aforementioned bound to compute error rates in the stable central limit theorem, when the entries are in the domain $\mathcal{D}_α$ of normal attraction of a stable law of exponent $α$. To conclude, we study the specific case where the entries are Pareto like multiplied by a slowly varying function, which provides an example of random variables that do not belong to $\mathcal{D}_α$, but for which our approach continues to apply.