Using a 2-quart saucepan, heat sugar and tap water to boiling over medium-high heat. Reduce heat to low, simmer, stirring until sugar dissolves. Remove from heat. Stir lemon juice and ice water into sugar syrup.
Pour lemonade into ten 5-ounce paper drinking cups. Place cups in shallow pan for easier handling; freeze pops until partially frozen, about 3 hours.
Insert wooden ice cream bar stick into center of each cup. Freeze until frozen solid, about 4 hours longer. To serve, peel off paper cups.