Using STEM ginger: Scrape the skin of stem ginger lightly with a small sharp knife, removing any blemishes as well as the papery bits of leaf sheath that are attached here and there. Trim away any dried ends. Rinse the pieces, pat them dry, and break them apart at the joints. Using a citrus zester or a grater with small sharp teeth, shred the ginger into threads, working over a bowl in order to catch all the juice. threads, working over a bowl in order to catch all the juice. You should have 1&1/3 to 1&1/2 cups, lightly packed.
Using MATURE ginger: Pare or scrape off the skin and grate the tender outer layer only (save the fibrous cores to make Fresh Ginger Jelly); grate enough to obtain 1&1/3 to 1&1/2 cups, lightly packed. Combine the grated ginger with 1 quart of water, bring to a boil, and boil 5 minutes. Drain; add fresh cold water and repeat the boiling and draining twice.
Combine the grated ginger, its juice (if you are using stem ginger), and 6 cups of water in a preserving pan; bring the mixture to a boil, cover, lower the heat, and simmer the ginger gently until the shreds are very tender and almost translucent, about 1&1/4 hours. Remove from the heat.
Measure the ginger and liquid. If you have more than 3 cups, spoon off the surplus liquid (save it--add sugar and mix the resulting syrup with chilled club soda for a pleasant refreshment while you make the marmalade).
Combine the measured ginger and liquid, the lemon juice, sugar, and salt in the rinsed-out preserving pan. Bring the mixture to a hard boil (a boil that cannot be stirred down) over medium-high heat and boil it exactly 1 minute. Stir in the liquid pectin and return to another full, rolling boil that cannot be stirred down; boil exactly 1 minute. Remove from the heat.
Stir the marmalade for 5 minutes to cool it slightly, then ladle it into hot, clean half-pint canning jars, leaving 1/4 inch of headspace. Seal the jars with new two-piece canning lids, according to manufacturer's directions and process the jars for 15 minutes in a boiling-water bath. Cool, label, and store the jars.