Watermelon Lemonade

This is slightly adapted from a recipe I found on Vocal Point.
1 seedless watermelon (about 7 pounds)
1/4 c sugar
1 lemon (squeezed for juice)
8 c ice cubes

Cut watermelon into chunks, removing the rind. Place 1/2 to 1/3 of it in a blender, with 1/2 to 1/3 of the sugar, and lemon juice and puree. Strain over a large pitcher. Repeat with remaining parts. Run ice through the blender, then add to the pitcher. Stir to mix in ice with the juice. Add more lemon juice or sugar to taste.