Put corned beef into large pot (or Dutch oven) and cover with water. Mix in the spice packet that came with the corned beef. Keeping the pot covered, bring it to a boil, then reduce to a simmer. Let simmer for about 50 minutes per pound (or until meat is tender).
Next, place whole potatoes and carrots into the pot. Cook until vegetables are almost tender. Add in the cabbage and cook for about 15 more minutes. Remove meat, and let the vegetables sit for 15 more minutes.
Finally, put vegetables in a bowl and cover. Add a healthy portion of broth (cooking liquid reserved in the Dutch oven or large pot) until amount is as desired. Cut the meat across the grain for best results.