Wash, peel, and chop the persimmons; add to a medium-sized pot. Cover with five cups of water and bring to a boil. Simmer for 20 minutes over low heat, until tender.
Drain all the liquid, then add the cinnamon, cardamom, and lemon zest.
Sprinkle with two cups of sugar, then heat over low to medium heat until the sugar begins to dissolve.
Stir frequently to prevent burning. Once the mixture comes to a boil, lower the heat and cook for an additional 30 minutes, stirring occasionally.
Add the lemon juice halfway through the cooking process. Mash with a potato masher to get the consistency that you prefer.
Remove from heat when the syrup is thickened. This should take approximately 30 minutes.
Ladle the preserves into two eight oz. sterilized mason jars. Run a knife along the edges of the filled jars to bring any bubbles that might have formed up to the surface. Tighten the lids on the jars and process in hot water for ten additional minutes.