An ordered semigroup $S$ is called a ${\it dual ordered semigroup}$ if $l(r(L)) = L$ for every left ideal $L$ of $S$ and $r(l(R)) = R$ for every right ideal $R$ of $S$ where $r(A)$ and $l(A)$ denoted the ${\it right annihilator}$ and the ${\it left annihilator}$ of a nonempty subset $A$ of $S$, respectively. The main result of this paper is to show the existence of 0-minimal ideals of a dual ordered semigroup.