Put currants and 1½ cups water into a heavy medium pot and bring to a boil over medium-high heat. Reduce heat to maintain a gently simmer and let berries cook, stirring occasionally with a wooden spoon, until berries are very soft, 15-20 minutes. Add sugar to berries and stir until dissolved. Increase heat to medium-high and bring jam to a vigorous boil, stirring often. Continue to boil, stirring frequently, until jam thickens and reaches its setting point (about 220º on a candy thermometer), 6-10 minutes. Remove pot from heat and skim any foam that has risen to the surface of the jam.
Meanwhile, submerge 2 pint canning jars, their lids and ring bands, and a widemouthed funnel into a large pot of boiling water over medium-high eat and sterilize for 10 minutes. Remove from hot water and transfer to a clean dish towel. Using the funnel, fill each jar with hot jam to no more than ¼" from the top. Wipe jar rims with a clean dish towel, place lids on jars, then screw on ring bands.
Transfer filled jars to a canning rack, submerge into pot of gently boiling water (jars should be covered by at least 1" of water), and place on a dish towel at least 1" apart to let cool undisturbed for 24 hours. To test that jars have properly sealed, press on center of each lid. Remove your finger; if lid stays down, its sealed. Refrigerate any jam that hasn't sealed and use within 4 weeks.