Beat yolks in a medium mixing bowl until thick and lemon colored. Gradually add sugar, beating well. Add rum, vanilla, and salt, beating well.
Soften gelatin in cold water in a saucepan. Bring to a boil, stirring until gelatin dissolves. Cool slightly. Slowly pour gelatin into rum mixture; beat constantly. Cool completely.
Beat whipping cream until soft peaks form. Fold into rum mixture. Pour into a lightly oiled 4-cup mold, and chill until set. Unmold pudding onto a serving platter. Spoon Blackberry Sauce over pudding.