Your permission to stay in the United States ends on the date shown on your I-94 Departure Record card—unless the card is marked “D/S” (duration of status). In that case, your permission to stay in the U.S. will expire on the program expiration date indicated in item 3 of your DS-2019.
If an extension is necessary, be sure to arrange it ahead of time by contacting your J-1 responsible officer at least two months before the expiration date shown on your DS-2019. You may have to bring proof of your source of funding and comprehensive insurance as well.