Cut the watermelon into wedges, then cut it away from the rind, cubing the flesh and picking out all the seeds.
Puree three quarters of the flesh in a food processor, but mash the last quarter on a plate, this will give the sherbet more texture.
Stir the sugar with the lemon juice and 1/2 cup of cold water in a saucepan over very low heat until the sugar dissolves and syrup clears.
Mix all the watermelon and the syrup in a large bowl and transfer to a freezer container.
Free for 1-1 1/2 hours, until the edges begin to set. Beat the mixture, return to the freezer for another hour.
When the hour is up, whisk the egg whites to soft peaks. Beat the ice mixture again and fold in egg whites. Return to the freezer for another, then beat once more and freeze firm.
Transfer the sherbet from the freezer to the fridge for 20-30 minutes before it is to be served.