Answer: Sounds like gray mold, which attacks strawberries when the weather is wet and humid, or the beds are overcrowded and the air circulation is poor. Thin your plants to give them lots of elbow room, make sure they're getting full sunshine all day, and pick off the infected berries as soon as you notice the mold. That should keep the problem in check. As the weather warms up, gray mold should be less of a problem.
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