Boil water. Pour over sugar in a bowl. Stir to dissolve. Cool and chill.
For a larger amount, the ratio is one kilo of sugar per litre of water (5 cups sugar and four cups water). Put these together in a saucepan and bring to a boil for 4 minutes to dissolve the sugar. Cool and chill.
Finely grate a tablespoon of zest from one washed grapefruit, and reserve. Combine the grapefruit juice, sugar syrup, lemon juice, and Campari in a jug. Chill in the fridge for a few hours until it is very, very cold.
Pour the sorbet mixture into an ice-cream machine and churn to slushy, 15 minutes. Whisk a spoonful or two with the egg white and zest, using a fork. Pour this back into the slushy sorbet and continue churning for five minutes. Spoon into a storage container and freeze a few hours so that the flavours intensify and the ice gets hard. Serve.