1 watermelon, seeded and pureed
1 cucumber, medium to large
2 tbsp lime juice
3/4-1 cup sugar
2 tsp chopped fresh basil
Peel the cucumber, chop into chunks and either run through a juicer or puree in a food processor (then strain through a mesh sieve). This should equal around 1/3 and 1/2 cup of juice. Add to watermelon puree, then add lime juice, sugar, and basil.
Put this combination in a jar with a lid and give it a really good shake to blend in the sugar. If you don't have a big enough jar, put part of the mixture in a smaller jar and shake, adding it back to the rest of the liquid. Chill in the fridge for at least an hour. Process in an ice cream maker according to the directions. Freeze the sorbet in a separate container for an hour or two before serving. Allow to thaw slightly before serving. Seems counter-intuitive to freeze then thaw, but I promise, this brings out more flavor. And, the basil will have had a chance to infuse a little more.
If you don't have an ice cream maker, you can survive without one. You'll just have to do a few more steps. Pour the mixture into a shallow pan with sides, cover, and put into the freezer. Every 20 minutes, scrape the mixture with a fork to loosen. Do this until you've reached the consistency of a sorbet, although it will be more like a granita with icy pieces. It will still be just as refreshing and delicious.