Wash the cucumber thoroughly and cut it into chunks without peeling.
Cut the melon in half, remove the seeds, and scoop out the flesh into cubes, or measure out 2 cups of pre-cut melon.
In a blender, combine the cucumber chunks, melon cubes, fresh lime juice, and honey or agave syrup.
Pour in 3 cups of cold water and blend on high until the mixture is smooth, about 1-2 minutes.
Taste the mixture and adjust sweetness if necessary by adding more honey or agave syrup.
Strain the mixture through a fine mesh sieve into a pitcher to remove pulp, if desired.
Chill in the refrigerator for 10 minutes, or serve immediately over ice.
Garnish with fresh mint leaves if desired, and pour into glasses.
For added flavor, include a few slices of ginger while blending, or serve as a slushy by blending with ice.
Store any leftover drink in the refrigerator for up to 2 days; shake well before serving as the ingredients may separate.