Can the United States give me a 1 Year J1 visa on a passport which?
Passports expire on their schedule, US visas expire on theirs. Your situation arises all the time, and is not a problem. The visa will still be valid even if/when the passport expires. You can travel to the US carrying a new passport plus the expired passport with the visa in it. As long as one of the passports is valid and the visa is valid, you are good to go.