  If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic
triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$,
$[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-\emph{hyperbolic}
$($in the Gromov sense$)$ if any side of $T$ is contained in a
$\delta$-neighborhood of the union of the two other sides, for every geodesic
triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta(X)$ the sharp
hyperbolicity constant of $X$, i.e. $\delta(X)=\inf\{\delta\ge 0: \, X \,
\text{ is $\delta$-hyperbolic}\}.$ In this paper we characterize the
lexicographic product of two graphs $G_1\circ G_2$ which are hyperbolic, in
terms of $G_1$ and $G_2$: the lexicographic product graph $G_1\circ G_2$ is
hyperbolic if and only if $G_1$ is hyperbolic, unless if $G_1$ is a trivial
graph (the graph with a single vertex); if $G_1$ is trivial, then $G_1\circ
G_2$ is hyperbolic if and only if $G_2$ is hyperbolic. In particular, we obtain
the sharp inequalities $\delta(G_1)\le \delta(G_1\circ G_2) \le \delta(G_1) +
3/2$ if $G_1$ is not a trivial graph, and we characterize the graphs for which
the second inequality is attained.
