In medium bowl, toss strawberries with sugar and lemon juice; let stand for 10 minutes.
Mash strawberries; set aside 1 cup (250 mL). Scrape remaining strawberry mixture into blender or food processor and purée until smooth. Strain through fine-mesh strainer.
Spoon ice cream into large bowl; fold in purée. Spoon half into airtight container; add half of the mashed berry mixture, swirling slightly with spoon. Repeat with remaining ice cream and berries. Refreeze until firm, at least 2 hours.