We will show that for $α>0$ there is $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices such that $αn< δ(G)< (n-1)/2$, then for every $n_1+n_2+\cdots +n_l= δ(G)$, $G$ contains a disjoint union of $C_{2n_1},C_{2n_2}, \dots, C_{2n_l}$ unless $G$ has a very specific structure.