In a large stock pot or very large dutch oven, brown the ground beef over medium-high heat. Once it starts to brown, add the chopped onion and cook until the beef is no longer pink. Drain the grease way. Return the meat and onion to the pot over medium heat. Stir in the garlic and cook about 1 minute. Add the undrained tomatoes, vegetable juice, and beef broth. Bring to a boil then stir in the rice. Reduce the heat to a simmer and add the cabbage. Stir well. Cook, covered, for 20 to 30 minutes, stirring occasionally, until the cabbage is tender and the rice is fully cooked. Generously add salt and pepper to taste. Add additional vegetable juice if you'd like a thinner soup.