Patients with Zollinger-Ellison syndrome require continuous control of their gastric acid secretion or else severe complications of peptic ulcer disease will occur. This study investigates the long-term efficacy of the oral gastric acid antisecretory drug, Omeprazole, which functions as a H+ - K+ ATPase inhibitor. Long-term safety will also be investigated. Also investigated is the ability of the parenteral H+-K+ ATPase inhibitor, pantoprazole to control acid secretion short-term, when oral Omeprazole cannot be used.