The DS-2019 controls your authorized status in the U.S. The visa is just for entry purposes, it does not determine your period of authorized stay, i.e. your lawful status in the U.S. You will have a 30 day grace period from that date to prepare for departure or apply for an extension or change of status. Check with your exchange program's Responsible Officer (RO) about possible extensions. J-1 visa holders who are subject to the home return requirement may not change status in the U.S. Those who are not subject to it may do so.
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