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.
Beat ice cream in a stand mixer with the paddle attachment just until smooth, about 30 seconds.
Scoop ice cream into a freezer-safe straight-sided container, alternating with blackberry mixture and chocolate.
Fold ice cream to form a swirl, pulling up from the sides and bottom. Press plastic wrap onto surface; cover. Freeze until firm.