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We consider an area-minimizing integral current $T$ of codimension higherthan $1$ in a smooth Riemannian manifold $\Sigma$. In a previous paper we havesubdivided the set of interior singular points with at least one flat tangentcone according to a real parameter, which we refer to as ``singularitydegree''. This parameter determines the infinitesimal order of contact at thepoint in question between the ``singular part'' of $T$ and its ``best regularapproximation''. In this paper we show that the set of points for which thesingularity degree is strictly larger than $1$, is $(m-2)$-rectifiable. In asubsequent 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.

The fine structure of the singular set of area-minimizing integral  currents II: rectifiability of flat singular points with singularity degree  larger than $1$