Why have the stems of one of my anemones rotted just above the soil?

This sounds like crown rot, a fungus disease that attacks some plants. Crown rot is especially likely when plants are set so low in the soil that water settles in and around the crowns instead of draining away. About all you can do once the rot starts is to dig up the plants, salvage the unaffected portions, and dip them in bichloride of mercury solution; then replant. To prevent crown rot, use sterilized soil and set the plant crowns higher than the surrounding soil.