Combine the sugar and berry juice in a pan and simmer over low heat while stirring until the sugar dissolves. Increase the heat slightly and simmer for 4 to 5 minutes. Cool for 10 minutes, then stir in the Créme de Cassis and lemon juice. Chill until cold.
Puree the frozen berries in a food processor and gradually add the cold sugar syrup. Add the egg white and blend for 1 minute. Pour the mixture into a freezer container, cover and freeze for 3 hours or until the sides begin to firm. Place the sorbet in a food processor and process to break up the ice crystals, then return to the container and freeze for a further 2 hours. Process the mixture in a food processor again and return to the freezer. Freeze for 3 to 4 hours until firm. Serve in scoops.