Cut the melon in half and scoop out and discard the seeds. Scoop out the melon flesh with a spoon and discard the shells.
Place the flesh in a food processor or blender with the icing sugar and lime or lemon juice and process to a purée. (Alternatively, rub through a sieve.) Pour into a freezer container, cover and freeze for 2–3 hours. If using an ice-cream maker, purée then pour into the machine, churn and freeze until half-frozen.
Whisk the melon mixture to break up the ice crystals, then whisk the egg white until stiff and whisk it into the half-frozen melon mixture. Return to the freezer until firm. Alternatively, add whisked egg white to the ice-cream machine and churn until very thick.
Transfer the sorbet to the fridge 20 minutes before serving to soften slightly or scoop straight from the ice-cream machine. Scoop the sorbet into glass dishes to serve. To make differently coloured sorbets, make up three batches of sorbet using a cantaloupe melon in one and honeydew and watermelon in the others.