Split plums in half, remove pits, and cut into 1 1/2-inch pieces. Do not peel. Place in a 6-quart nonreactive saucepan, and add 1 cup water and the sugar. Bring to a boil, reduce heat, and cook until fruit is very soft, about 20 minutes. Remove from heat; puree fruit and liquid in a food mill or food processor.
Rinse saucepan and return puree to pan along with cardamom and cinnamon. Cook over low heat, stirring frequently to prevent scorching, until thick enough to spread, 2 1/2 to 3 hours. Remove from heat and let cool. Store in an airtight container.
The butter will keep for up to 1 month in the refrigerator, 6 months in the freezer.