Purée berries and 1 cup ice cream in blender until smooth. Strain through a fine sieve, pressing on berry mixture to extract liquid. Discard solids. Transfer berry mixture to a clean blender with remaining 3 cups ice cream, and purée until smooth.
Divide mixture among sixteen 2 12/2 oz. ice-pop molds, and freeze until hardened, at least 4 hours. pops can be frozen for up to 3 days.