Cut orange in half crosswise; remove and discard seeds. Clip membrane, and carefully remove pulp. Set pulp aside. Drain orange shells upside down on paper towels.
Position knife blade in food processor bowl; add orange pulp. Process until smooth. Pour puree into a wire-mesh strainer; press with back of spoon against the sides of the strainer to squeeze out juice. Discard pulp remaining in strainer. Combine strained juice, 1/4 cup orange juice, and sugar, stirring well. Pour mixture into an 8-inch square pan. Cover and freeze at least 8 hours or until firm.
Position knife blade in food processor bowl; add frozen orange mixture. Process until smooth but not thawed. To serve, scoop into 2 dessert dishes or into orange shells, if desired. Serve immediately.