1. Stir together strawberries and sugar in a medium bowl. Let stand 30 minutes, stirring occasionally. Pulse strawberry mixture, basil, and lime zest in a food processor 9 to 10 times or until berries are finely chopped (almost pureed), stopping to scrape down sides as needed.
2. Whisk together yogurt and cream in a medium bowl until smooth; stir strawberry mixture into yogurt mixture until well blended. Cover and chill 1 hour.
3. Pour strawberry mixture into freezer container of a 1 1/2-qt. electric ice-cream maker, and freeze according to manufacturer's instructions. (Instructions and times may vary.)