In a blender combine lime pieces, 2 cups water, and 2/3 cup sugar. Cover and blend about 30 seconds or until limes are chopped; do not puree. Strain through a sieve into a large pitcher. Mix in 2 cups additional water. Cover and chill for 1 to 12 hours.

Stir rum into lime mixture. Moisten the rims of 8 glasses with the reserved lime piece; dip rims in sugar. Divide mint among glasses. Use a wooden spoon to press mint into the bottom of each glass; press firmly to release flavor. Fill glasses with ice cubes. Pour chilled lime mixture over ice and mint leaves. If desired, add extra quartered limes and/or mint leaves.