Boil up grape juice, sugar, honey and lemon juice. Cut melons in half, remove seeds with a spoon and add to the sugar syrup. Let them cool down, pour through a sieve and collect the syrup. Cut out balls from 2 melon halves. Remove the remaining pulp and the remaining half of the melon from the skin. Wash basil leaves, except for a few for garnishing, dab dry and cut into strips.
Puree the melon, sugar syrup and basil and place in a bowl. Cover and place in the freezer and freeze for about 6 hours. Stir with a fork every 15-30 minutes. Put melon balls into glasses. Stir the sorbet mixture again (if necessary, let it thaw a little) and shape it into balls with an ice-cream cone former. Also distribute into the glasses and serve garnished with basil