  We consider integral area-minimizing $2$-dimensional currents $T$ in
$U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![\Gamma]\!]$, where $Q\in
\mathbb N \setminus \{0\}$ 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 $\Omega\subset \mathbb R^{2+n}$ is a bounded uniformly
convex set and $\Gamma\subset \partial \Omega$ a smooth $1$-dimensional closed
submanifold, then any area-minimizing current $T$ with $\partial T = Q
[\![\Gamma]\!]$ is regular in a neighborhood of $\Gamma$.
