  We consider an area-minimizing integral current $T$ of codimension higher
than $1$ in a smooth Riemannian manifold $\Sigma$. In a previous paper we have
subdivided the set of interior singular points with at least one flat tangent
cone according to a real parameter, which we refer to as ``singularity
degree''. This parameter determines the infinitesimal order of contact at the
point in question between the ``singular part'' of $T$ and its ``best regular
approximation''. In this paper we show that the set of points for which the
singularity degree is strictly larger than $1$, is $(m-2)$-rectifiable. In a
subsequent work we prove that the remaining flat singular points form an
$(m-2)$-null set, thus concluding that the singular set of $T$ is
$m-2$-rectifiable.
