Boil the beets in a large saucepan with four cups of water until they are pieced easily with a fork, about 30 – 40 minutes. Remove from heat. Strain the cooking liquid through a fine mesh strainer and reserve. Store the beets in the refrigerator for another use.
Add enough water to the liquid so you have 3 ½ cups. Stir in the vinegar, sugar, lemon juice and orange juice. Taste and correct the seasonings. The taste should be a refreshing sweet and sour. Cool the liquid. Freeze in an ice-cream maker according to the manufacturer’s directions.
Nutritional analysis per serving: 57 calories, less than 1 gram protein, 14 grams carbohydrates, less than 1 gram fat, 6 mg. sodium, less than 1 gram fiber.