In a graph $G$, a vertex dominates itself and its neighbours. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The minimum cardinality among all double dominating sets of $G$ is the double domination number. In this article, we obtain tight bounds and closed formulas for the double domination number of lexicographic product graphs $G\circ H$ in terms of invariants of the factor graphs $G$ and $H$.