This study will employ the Vessix Renal Denervation System (Boston Scientific) to ablate the renal nerve by radio-ablation in patients suffering from severe pain due to Autosomal Dominant Polycystic Kidney Disease (ADPKD). Twenty patients with opiate-dependent, Autosomal Dominant Polycystic Kidney Disease-associated kidney pain will be selected to this double-blind, prospective device study. Subjects will be randomly assigned to an experimental (those who will receive nerve ablation) or control (those who will have a sham procedure in which the ablation device is not turned on) groups. Subjects will complete physical exams and be administered questionnaires regarding their mental status and pain before, during and after the denervation procedure. Renal function and adverse events will be monitored for 1 year after denervation.