Bring the mixture to a boil, stirring until the sugar is dissolved; boil until it registers 220F on a candy thermometer.
Stir in the pineapple and the ginger; simmer the mixture, uncovered, stirring to prevent scorching for 1 hour, or until it registers 220 F on a candy thermometer.
Remove the kettle from the heat and ladle the preserves into 5 sterilized 1/2-pint jars, filling the jars to within 1/4 inch of the tops.
Wipe the rims with a dampened towel and seal the jars with the lids.
Screw on bands finger-tight.
Put the jars in a water bath canner or on a rack set in a deep kettle.
Add enough hot water to the canner or kettle to cover the jars by 2 inches, and bring it to a boil.
Process the jars, covered, for 10 minutes.
Transfer them with tongs to a rack or countertop protected with a heat-resistant mat, and let them cool completely.