Rinse and dry the currants, then remove any remnants of the vines. Add to a pot and crush with a fork. Mix in the jam sugar and tarragon sprig. Let marinate together for 30 minutes. Bring to a boil, and cook for 6-8 minutes. Remove from the heat, and remove the tarragon sprig.
Divide the jam into the hot, rinsed jars. Fill to the brim, then seal well. Turn upside down for 5 minutes, then turn right side up and cool completely.