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Add the yogurt and honey to the chilled syrup and stir until thoroughly combined. Put a bit of the yogurt mixture into each of the molds, to a height of about 3/4 inch. Freeze until the mixture begins to set, about 40 minutes.
If the blackberries are big, cut them in half. Divide the blackberries among the molds, then pour in the remaining yogurt mixture, dividing it evenly among the molds.
If using conventional molds, snap on the lid and freeze until solid, 3 to 4 hours. If using glasses or other unconventional molds, freeze until the pops are beginning to set (45 minutes to 1 hour), then insert the sticks and freeze until solid, 3 to 4 hours. If using an instant ice pop maker, gently fold the blackberries into the yogurt prior to filling the molds and follow the manufacturer's instructions.