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The Indians would dry the fruits and save them like prunes. The genus name is a derivative of greek and means "Fruit of the God Zeus." Apparently someone really liked them!
At my in-laws house there is a tall persimmon tree just off the driveway which makes it perfect for seed collecting. The fruits fall to the ground and dry up on the driveway leaving behind the little seeds. Persimmon seeds need a period of cold followed by warmth to germinate called stratification. There are two ways that I stratify seeds. The first way is to place the seeds in mixture of peat and vermiculite or peat and sand then place it all into a plastic bag that can be sealed. Moisten the soil enough so that it is damp but not soaked then close the bag and store it in the refrigerator until spring. Then plant it in the ground. The second way (which is much easier) is to plant the seeds in a pot stored outdoors. In the spring check for germination and transplant anything that has grown into its own pot then leave the pot alone until next year. This is what I just did for some Japanese maple seeds. I used the refrigerator method for redbuds last year and did get one to germinate. I'll use the second method with these seed just to see if I can get any to grow.
Winter Garden Update: It's planted, I'll tell you what I did very soon! Also if you drew a plan up for the Winter Gardening Design Post let me know and I'll can link to it!