Peel and core the apples and boil them with the sugar and 2 tablespoons of water until soft. If necessary, drain, then puree and let cool.
Whip the cream until stiff and fold in the cooled apple mixture. Pour the mass into a plastic container and freeze. Beat the half-frozen ice cream again and refreeze it until it is firm.
Put the ice cream in the fridge for 30 minutes before serving so that it thaws a little.
Puree the blackberries with 6 teaspoons of water and the sugar in the food processor. Press through a sieve, then bring to the boil in a saucepan and let simmer for 1 minute. The sauce should be clear and shiny.
Serve the blackberry sauce with the apple ice cream.