We consider integral area-minimizing 2-dimensional currents $T$ in $U\subset \mathbf {R}^{2+n}$ with $\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$ , where $Q\in \mathbf {N} \setminus \{0\}$ and $\Gamma $ is sufficiently smooth. We prove that, if $q\in \Gamma $ is a point where the density of $T$ is strictly below $\frac{Q+1}{2}$ , then the current is regular at $q$ . The regularity is understood in the following sense: there is a neighborhood of $q$ in which $T$ consists of a finite number of regular minimal submanifolds meeting transversally at $\Gamma $ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for $Q=1$ . As a corollary, if $\Omega \subset \mathbf {R}^{2+n}$ is a bounded uniformly convex set and $\Gamma \subset \partial \Omega $ a smooth 1-dimensional closed submanifold, then any area-minimizing current $T$ with $\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$ is regular in a neighborhood of $\Gamma $ .
