Place the chopped melon in a blender jar. Reserve a ¼ of the cucumber; roughly chop the remainder and add it to the blender along with the basil, lime juice and salt. Puree until completely smooth. Chill or serve immediately. (see note)
Cut the remaining cucumber in ¼-inch dice. Divide the soup into bowls. Garnish with some diced cucumber, a drizzle of olive oil, a pinch of the smoked paprika and if desired, a few grains of high quality salt.
Add a couple of ice cubes to the blender for insured coolness.