  We consider an area-minimizing integral current $T$ of codimension higher
than 1 ins a smooth Riemannian manifold $\Sigma$. We prove that $T$ has a
unique tangent cone, which is a superposition of planes, at
$\mathcal{H}^{m-2}$-a.e. point in its support. In combination with works of the
first and third authors, we conclude that the singular set of $T$ is countably
$(m-2)$-rectifiable.
