A

连接$BD$.

$\because AB=BC$，$\angle ABC=120^{\circ}$，
$\therefore \angle A=\angle C=\frac{1}{2}(180^{\circ}-\angle ABC)=30^{\circ}$，
$\because AB$的垂直平分线是$DE$，
$\therefore AD=BD$，
$\therefore \angle ABD=\angle A=30^{\circ}$，
$\therefore \angle CBD=120^{\circ}-30^{\circ}=90^{\circ}$，
$\therefore DC=2BD$，
$\therefore DC=2AD$，
$\because AC=6$，
$\therefore AD=\frac{1}{3}\times 6=2$，
故选：$A$.