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Let $X$ be a compact K\"ahler manifold and $\a \in H^{1,1}(X,\R)$ a K\"ahlerclass. We study the metric completion of the space $\HH_\a$ of K\"ahler metricsin $\a$, when endowed with the Mabuchi $L^2$-metric $d$.  Using recent ideas of Darvas, we show that the metric completion$(\overline{\HH}_\a,d)$ of $(\HH_\a,d)$ is a CAT(0) space which can beidentified with $\E^2(\a)$, a subset of the class $\E^1(\a)$ of positive closedcurrents with finite energy.  We further prove, in the toric setting, that$\overline{\HH}_{\a,tor}=\E_{tor}^2(\a)$.

The metric completion of the Riemannian space of K\"{a}hler metrics