Rinse ribs and pat dry. Trim and discard excess fat.
Prepare barbecue for indirect heat (see below). When grill is medium-hot (you can hold your hand at grill level only 3 to 4 seconds), lay ribs on it and turn as needed until well browned, 40 to 50 minutes.
Baste 1 side of ribs with half the sauce. Turn ribs sauce side down and cook until sauce browns and forms a thick, sticky glaze, about 10 minutes. Baste top of ribs with remaining sauce, turn over, and cook until sauce browns and forms a thick, sticky glaze, about 10 minutes longer.
Transfer ribs to a platter and garnish with blackberries and mint. Cut between bones to separate portions.
To prepare grill for indirect heat:
If using charcoal briquets, mound and ignite 60 briquets on the firegrate of a barbecue with a lid. When the briquets are hot, after 15 to 20 minutes, push equal amounts to opposite sides of the firegrate. Add 5 more briquets to each mound of coals now and every 30 minutes while cooking. Set a drip pan on the firegrate between the coals. Set the grill in place.
If using a gas barbecue, turn all the burners to high, close the lid, and heat for 10 minutes. Then adjust the burners for indirect cooking (heat on opposite sides of grill, not down center under food), and keep on high unless recipe specifies otherwise.