Combine chicken, water, chopped celery stalk, salt, and pepper in a large Dutch oven. Bring to a boil; cover and simmer 1 hour or until chicken is tender. Remove chicken from broth; cool. Bone chicken, and cut into 1-inch cubes. Strain broth, and return broth to Dutch oven. Discard celery.
Add next 4 ingredients to broth; cook over low heat 15 minutes. Add cabbage; cook an additional 5 minutes. Stir in chicken, and bring mixture to a boil. Serve immediately.