Place watermelon cubes in a blender. Blend until smooth.

In a small pitcher, combine fresh mint leaves, sugar, and fresh lime juice. Using the handle of a wooden spoon, muddle the ingredients to release the oils from the mint. Take care not to muddle to the point where the leaves fall apart. Pour rum and the watermelon puree into the pitcher and stir gently.

Fill four tall glasses (such as Collins glasses) with ice and divide the mojito amongst the glasses. Garnish with lime slices and mint sprigs, and serve immediately.