We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent $α+1$, where $α\in (1,2)$. The limiting components are constructed from random $\mathbb{R}$-trees encoded by the excursions above its running infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive $α$-stable Lévy process. These spanning $\mathbb{R}$-trees are measure-changed $α$-stable trees. In each such $\mathbb{R}$-tree, we make a random number of vertex-identifications, whose locations are determined by an auxiliary Poisson process. This generalises results which were already known in the case where the degree distribution has a finite third moment (a model which lies in the same universality class as the Erdős--Rényi random graph) and where the role of the $α$-stable Lévy process is played by a Brownian motion.