Stir the lemon juice and sugar together in a pitcher or mixing bowl until the sugar dissolves.
Stir in the cold water, then refrigerate the lemonade for at least 1 hour.
Add 1 cup of the prepared lemonade and 2 cups of the ice cream to a blender.
Blend until smooth.
Pour into a glass and serve immediately with lemon slices, if desired.
Notes
You can use bottled lemon juice for this recipe. If using bottled juice, increase the amount to 1 ½ cups of lemon juice. This drink is best served immediately. You can store it in the freezer covered for up to 1-2 days.