In a small saucepan, bring sugar and water to a boil. Reduce heat; simmer 10 minutes. Remove from heat; add rosemary. Steep, covered, 10-15 minutes according to taste. Discard rosemary. Cool completely. Store in an airtight container in the refrigerator up to 1 month.
For each serving, in a blender, combine frozen strawberries, rum, lime juice and 2 tablespoons rosemary syrup; cover and process until smooth. Pour into a chilled glass; garnish with a whole strawberry and additional rosemary sprig.
