Cut the melon in half and remove and discard the seeds, then roughly chop the flesh – you need about 450 g (14½ oz). Place in a food processor with the sugar, ginger and lime juice, then blend until smooth.
Transfer the sorbet to an ice cream maker and process according to the manufacturer's instructions. If you don't have an ice cream maker, place the mixture in a freezer-proof container and freeze for about 2–3 hours or until ice crystals have appeared on the surface. Beat with a hand-held electric whisk until smooth, then return to the freezer. Repeat this process twice more until you have a fine-textured sorbet and freeze until firm.
Remove the sorbet from the freezer 10 minutes before serving. Serve, scooped into glasses with a wafer.