Bring sugar and 1 cup water to a boil in a saucepan, stirring to dissolve sugar. Stir in ginger. Remove from heat. Let cool.
Transfer ginger pieces to a blender with remaining 21/2 cups water, and blend until finely chopped. Strain, discarding solids. Pour ginger water into syrup, and stir in lemon juice. Pour into a 9-by-13-inch metal baking pan, and freeze overnight.
Break frozen mixture into chunks, and transfer to a bowl. Whisk until slushy. Divide among glasses, and garnish with lemon rounds.
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