Brown the ground beef in a skillet over medium heat until no pink remains. Drain any excess fat and add the beef to the slow cooker.
Add the chopped cabbage, diced onion, sliced carrots, minced garlic, diced tomatoes, beef broth, rice, paprika, salt, and black pepper to the slow cooker.
Stir everything together until the ingredients are evenly mixed.
Cover and cook on low for 7 hours, until the cabbage is tender and the rice is cooked through.
Ladle the soup into bowls and top each serving with a dollop of sour cream.