1. Drain the currants and pour the syrup into a saucepan with the sugar. Bring to a boil and reduce over high heat by a third.
2. Puree the currants in a food processor. Pass the fruit puree through a medium sieve, then through a fine sieve. Discard all skins and seeds.
3. Combine the sieved pulp, creme de cassis, and syrup. Chill thoroughly.
4. Put this mixture in an ice cream machine and proceed according to the manufacturer's directions.