In a large, heavy kettle or Dutch oven, combine apples and cider. Cover and cook until soft, about 15 minutes; adding more liquid if necessary.
Press mixture through a fine sieve or food mill. Return puree to kettle. Add sugars, lemon juice, cinnamon, ginger and salt. Bring to a boil. Reduce heat and simmer, uncovered, until thickened and no more liquid separates from pulp, about 45-60 minutes, stirring occasionally.
Ladle into hot, sterilized canning jars, leaving 1/4-inch head space. Place lids on jars and tighten screw bands. Place jars on a rack in a pot of simmering water, making sure jars are completely covered by water. Bring to a rolling boil. Boil for 10 minutes; cool. Gradually remove jars from water. Check for proper seal.
Makes 5 half-pints.
