Combine the sugar in a saucepan with 1 pint cold water. Place over a low heat and stir until the sugar dissolves.
Once the liquid is clear, raise the heat under the pan and bring to a boil. Simmer steadily for 5 minutes, in order to make a syrup.
Meanwhile, peel the pink grapefruit, removing as much white pith from around the flesh as possible. Working over a mixing bowl to catch the juices from the fruit, cut the flesh into segments by running a serrated fruit knife down each side of the membrane dividing the fruit's segments.
Ease the cut segment out into the mixing bowl and, when each segment has been cut away, squeeze the remaining membrane to extract as much juice as possible before discarding it.
Pick out any seeds.
Tear the basil leaves into tiny pieces and stir them into the sugar syrup. Pour the syrup over the grapefruit, stir gently to coat evenly, then cover and marinate in the fridge for about 2 hours before serving.