In a blender, combine the chopped cucumber, 1 cup of cold water, lime juice, mint leaves, and sugar.
Blend until completely smooth.
Strain the mixture through a fine-mesh sieve into a large pitcher to remove the pulp.
Stir in the remaining 2 cups of cold water.
Taste and adjust sweetness if needed.
Serve over ice and garnish with mint sprigs or cucumber slices, if desired.