Summary: A graph \(G(V, E)\) is claw-free if no vertex has three pairwise non-adjacent neighbours. The maximum weight stable set (MWSS) problem in a claw-free graph is a natural generalization of the matching problem and has been shown to be polynomially solvable by Minty and Sbihi in 1980. In a remarkable paper, Y. Faenza et al. [J. ACM 61, No. 4, Article No. 20, 41 p. (2014; Zbl 1321.05260)] have shown that, in a two-step procedure, a claw-free graph can be first turned into a quasi-line graph by removing strips containing all the irregular nodes and then decomposed into {claw, net}-free strips and strips with stability number at most three. Through this decomposition, the MWSS problem can be solved in \(\mathcal{O} (|V|(|V| \log |V| + |E|))\) time. In this paper, we describe a direct decomposition of a claw-free graph into {claw, net}-free strips and strips with stability number at most three which can be performed in \(\mathcal{O}(|V|^2)\) time. In two companion papers we showed that the MWSS problem can be solved in \(\mathcal{O}(|E| \log |V|)\) time in claw-free graphs with \(\alpha (G) \le 3\) and in \(\mathcal{O}(|V| \sqrt{|E|})\) time in {claw, net}-free graphs with \(\alpha (G) \ge 4\). These results prove that the MWSS problem in a claw-free graph can be solved in \(\mathcal{O}(|V|^2 \log |V|)\) time, the same complexity of the best and long standing algorithm for the MWSS problem in line graphs.