1. Mash berries with a fork in a microwave-safe bowl. Stir in sugar. Microwave, stirring every 20 seconds, until sugar has dissolved, 1 minute. Let cool completely.
2. Beat ice cream in a stand mixer with the paddle attachment just until smooth, about 30 seconds.
3. Scoop ice cream into a freezer-safe straight-sided container, alternating with blackberry mixture and chocolate.
4. Fold ice cream to form a swirl, pulling up from the sides and bottom. Press plastic wrap onto surface; cover. Freeze until firm.