Brown the duck in the butter, then transfer it to a heavy Dutch oven. Pour off most of the fat from the skillet and add the onions and garlic. Cook, stirring, until onions are wilted.
Add the hot stock to the skillet. Stir with a wooden spoon to dissolve the brown particles which cling to the bottom and sides of the skillet.
Pour the liquid over the duck and add salt and pepper to taste; cover. Cook for about one hour, until duck is almost tender. Remove excess fat. Add the remaining ingredients, cover, and cook for twenty minutes, or until rice is tender.