Dice your watermelon, and place the pieces in the freezer overnight. After several hours in the freezer, you can proceed, but if the watermelon isn't completely frozen, the sorbet will have more of a slushy texture rather than be a solid sorbet.
Place your diced, frozen watermelon into the food processor and add some lime juice. I used 2-3 cups of watermelon for each half lime, but it's a good idea to just add a little at the beginning and add more, to taste, later on.
Don't add too much liquid at the beginning or it will tend to fly out of your food processor. (Don't ask me how I know that!) Start with just a few drops, and slowly add in more as you process your sorbet.
Continue to process your watermelon and lime juice in your food processor until you get a sorbet-like texture. You can add a little more lime juice as needed for taste and texture.
If you want a sweeter sorbet, you can add in a little honey or another sweetener of your choice. I usually just leave it plain as I prefer it without. I have tried adding honey, and it works OK, but there were a few places where the honey froze up into small drops within the sorbet. It didn't really bother me, but it's something to consider before adding it because it does change the texture somewhat. Adding liquid stevia extract doesn't usually change the texture, but I don't like the flavor as much.
If you'd prefer to sweeten this with a simple sugar syrup, as is usually done, you can make one by dissolving equal parts of sugar and water and cooking them for a few minutes over low to medium heat. Once cool, you can add the syrup to the food processor with your watermelon chunks.