What are you using them for, Jen? If you're making jelly, don't worry about the stems. In a pot, pour enough water over the berries just to cover. Boil until they break down, and run through food mill. Then you'll have juice for your jelly.
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The only time I ever had mulberries, they were so full of seeds that I couldn't stand them. I don't even like to eat blackberries that often because of the seeds (I love blackberries for their flavor though). I'm very strange about seeds!