Put the watermelon chunks in a blender and blend until smooth. Strain through a fine strainer, pressing through well but holding back seeds and thick pulp. You should get around 4oz/120ml of juice.
Squeeze the juice from the lime into a hi-ball glass and add the mint leaves and sugar - muddle together with a muddler (or pestle/end of wooden spoon if you don't have one).
Add the watermelon juice and rum, mix together, then add some ice. Top up the glass with soda water and stir before serving. Slap a sprig of mint in your hand to use as a garnish, as well as a slice of watermelon over the glass.
Notes
I've suggested 1 ½oz rum for one drink, but you can also make it with 2oz for a bit stronger. You could also reduce the watermelon juice but I think it works well as a slightly longer drink, as written here.