In a blender, whirl 3 1/2 cups rinsed, drained blackberries until smooth. Push through a fine strainer into a 1-quart glass measure; discard seeds. Add 1/2 cup apple juice, 2 to 4 tablespoons sugar (to taste), and 1/8 teaspoon ground cardamom to purée; stir until sugar is dissolved.
Pour fruit mixture into 6 to 8 juice bar molds (see notes). Attach covers firmly and insert sticks, leaving 1 1/2 to 2 inches of each sticking out. Set molds in freezer, making sure they're level and upright, and freeze until firm, at least 3 hours, or up to 2 weeks.
To unmold, run warm water over the molds up to the rim, just until pops are released from sides, 5 to 15 seconds. Remove the covers and pull out the pops.
Layered Pops: To create two-tone pops, make two different recipes and use twice as many molds. Pour one mixture into all the molds and freeze until firm to the touch on top, about 45 minutes, then pour the second mixture over the first, and freeze completely.