Classification of connected Hopf algebras of dimension $p^3$ II

Let $k$ be an algebraically closed field of prime characteristic $p>0$. In a preceding paper arXiv:1309.0286, we provided the classification of connected Hopf algebras over $k$ of dimension $p^3$, except for the case when the primitive space of the Hopf algebra is a two-dimensional abelian restricted Lie algebra. In this subsequent paper, we complete the classification by considering the remaining case, for $p>2$.