Bring water and sugar to a boil in a small saucepan. Stir until the sugar is completely dissolved, scraping down the sides as it boils. Remove syrup from the heat and let cool. Put in an airtight container and refrigerate for 24 hours.
Puree the pulp to a smooth consistency. Mix with the lime juice.
Strip the leaves off the verbena and chop them finely. Mix with the pulp. Refrigerate for 24 hours.
Strain the pulp through a fine sieve. Use a silicone spatula to squeeze out as much liquid as possible.
Measure 8 ounces of syrup and 12 ounces of melon liquid. Mix well. Process in an ice-cream maker according to manufacturer’s instructions.
Fill sorbet in an airtight freezer container and freeze for 12 hours, or until solidly frozen. Take sorbet out of the freezer 15 to 20 minutes before serving to soften, but not much longer because it melts quickly.