Rinse the corned meat, put it in a pot along with the salt pork, cover with water, and boil for 10 minutes. Pour off the water, cover again with cold water, and bring to a new boil. Add the bay leaves, onions, pepper, and basil. Cover tightly and simmer (but do not boil) for 5 or 6 hours, or until fork-tender. Add the potatoes, carrots, and turnips; cover and cook until the potatoes are done. Add the cabbage, cover tightly, and simmer for 10 minutes. Note that it is not necessary to cover the cabbage with liquid; if you cover the pot tightly, the steam will cook it.
Center the corned beef on a large platter and arrange the vegetables around it in groups. Serve hot with corn bread.