Answer: First, some general information: Impatiens are susceptible to what is commonly called stem rot. This disease can be caused by a number of organisms, including several fungi and at least one bacteria. These pests dwell in the soil; when they infect a plant, they girdle the stem, causing the area above the rot to wilt. Stem rot is generally worse in heavy, wet soils, so plant impatiens only in areas where soil is well-drained, or plant them in containers.
Rhizoctonia is one of the possible culprits; this is also one of the diseases that can cause damping off of young seedlings. It is indeed possible that the rain splashed spores onto the plant -- Rhizoctonia (and many other soil fungi) are quite widespread. The fungi often enters a plant through a wound -- so if the heavy rains caused stems to crack and break, the fungi could have entered there.
Since the organisms are so widespread, the best way to minimize their damage is to maintain healthy, vigorously growing plants, and make sure the soil they are in drains well. A layer of mulch can help prevent spores from splashing up onto the plant; however, it can also encourage disease because it keeps the area around the stems moister. I'd mulch between plants, but leave a section around each stem unmulched. If you find you have a big problem with this disease, you might want to try growing your impatiens in containers filled with sterile potting soil.
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