Spanning subdivisions in Dirac graphs

We show that for every $n\in \mathbb{N}$ and $\log n\le d\le n$, if a graph $G$ has $N=\mathrm{\Theta }(dn)$ vertices and minimum degree $(1+o(1))\frac{N}{2}$, then it contains a spanning subdivision of every $n$-vertex $d$-regular graph.