Let $T$ be an area-minimizing integral $2$-current in an open subset $U \subset \mathbb R^3$. For every point $x_0 \in \operatorname{spt} T \setminus \operatorname{spt} \partial T$, there exist $r>0$, an integer $q \in \mathbb N$, and a smooth embedded minimal surface $\Sigma \subset B(x_0,r)$ such that $\partial \Sigma \cap B(x_0,r)=\varnothing$ and $T\llcorner B(x_0,r)=q[[\Sigma]]$. Hence away from $\operatorname{spt}\partial T$, the support of $T$ is a smooth minimal surface, with locally constant integer multiplicity on each connected regular sheet.