In a Dutch oven, place the corned beef brisket and cover it with water. Add the spice packet, cover the pot, and bring it to a boil over high heat.
Reduce the heat to low and let the corned beef simmer for about 2 hours, or until it's almost fork-tender.
While the corned beef is cooking, prepare the vegetables. Cut the potatoes in half, peel and cut the carrots, and cut the cabbage into small wedges.
After 2 hours, add the potatoes and carrots to the Dutch oven. Continue cooking until the vegetables are almost tender and the meat is fork-tender, about 10 minutes.
Finally, add the cabbage to the pot and continue cooking until it's tender, about 15 more minutes.
Remove the corned beef from the pot and let it rest for 10 minutes. Slice the meat against the grain and serve it with the vegetables.