Make sure all of your tools are very clean. I like to run the dishwasher with the bowl, knife, cutting board, and canning jars to be sure they are sterile, but hot soapy water does the trick, too. Rinse well and air dry.
Remove the wilted outer leaves, setting one aside to use later.
Chop the cabbage. Cut it in quarters, then remove the core. Cut the quarters in half again, and then cut very thin crossways strips. Place in a large bowl.
Sprinkle the salt on the cabbage, and then using your clean hands squeeze and crush the cabbage until it is limp and juicy. This takes about 8-10 minutes. You want quite a bit of the juice to be released.
Add the caraway seeds, if desired, at this time.
Pack the cabbage into the quart jar. Press it down with your fist. When it's all in the jar, pour in the cabbage juice and then put one of the outer leaves on top and press down again to submerge all of the chopped cabbage into the liquid.
Place the smaller, weighted jar on top of the cabbage and press it down again. Check on it and press down again every couple of hours during the first 24 hours.
Leave the jar in a room temperature place (65 to 75 degrees) and away from direct sunlight. Check every day to make sure the cabbage is still immersed. If the cabbage doesn't stay under the liquid, add a little bit of salt water (dissolve 1/2 tsp salt in 1/2 cup water).
After about 3 days, taste to see if the cabbage is fermented to your liking. You can continue the fermentation process for up to about 10 days.
When it tastes right to you, remove the weighted jar and the outer cabbage leaf, put the lid on the jar, and refrigerate. It will keep for several months in the refrigerator.