1. Combine first 4 ingredients; beat with a mixer until smooth. Beat in crème fraîche and 1 cup milk. Stir in 3 cups milk and corn syrup. Pour into the freezer can of an ice-cream freezer; freeze according to manufacturer's instructions.
2. Combine blackberries and lemon juice in a food processor; process until smooth. Strain the berry mixture through a fine sieve over a bowl; discard solids. Scrape ice cream into a freezer-safe container. Drizzle berry mixture over top of ice cream; swirl gently. Cover and freeze 1 hour.