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In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory $\gamma$ between two points on the boundary of a finite subdomain of ${\mathbb{Z}}^d$ is proportional to ${\mu }^{-\text{length}\left(\gamma \right)}$. When $\mu$ is supercritical (i.e. $\mu lt;{\mu }_c$ where ${\mu }_c$ is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.