This randomized, open-label, multicenter, international Phase IIIb study will compare the efficacy and safety of two Herceptin dosing regimens in combination with cisplatin/capecitabine chemotherapy in participants with HER2-positive metastatic gastric or gastro-esophageal junction adenocarcinoma. Participants who have not received prior treatment for metastatic disease will be randomized to receive Herceptin intravenously as either an 8-milligram per kilogram (mg/kg) loading dose followed by 6 mg/kg every 3 weeks (q3w) as standard of care or an 8-mg/kg loading dose followed by 10 mg/kg q3w until disease progression. Capecitabine will be administered for 6 cycles at a dose of 800 milligrams per meter-squared (mg/m^2) orally twice a day on Days 1 to 14 of each 3-week cycle, and cisplatin will be administered intravenously for 6 cycles at a dose of 80 mg/m^2 on Day 1 of each 3-week cycle. Herceptin will be continued until disease progression occurs.