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

The Fine Structure of the Singular Set of Area-Minimizing Integral  Currents III: Frequency 1 Flat Singular Points and $\mathcal{H}^{m-2}$-a.e.  Uniqueness of Tangent Cones