In a medium saucepan combine sugar and water. Cook and stir over high heat until mixture comes to a boil and sugar dissolves. Remove from heat; cool.
In a blender container combine half of the sugar mixture and berries; cover and blend until almost smooth. Pour into a fine mesh sieve set over a bowl. Repeat with remaining sugar mixture and berries. Press berry mixture through sieve; discard seeds. Transfer berry mixture to a large mixing bowl. Stir in yogurt an vanilla; mix until well combined. Cover and chill several hours or overnight.
Freeze mixture in a 2-quart ice-cream freezer according to manufacturers directions. Ripen 4 hours. Cut melon or other fruit into long, thin ribbons with a vegetable peeler, if desired. Serve with scoops of frozen yogurt. Makes 1-1/2 quarts (6 to 8 servings).