Melt the butter in a Dutch oven or large heavy-bottomed pot. Add the whole potatoes, salt, and pepper, and toss well. Cover the pot tightly and cook over low heat for 20 to 30 minutes, until the potatoes are just tender when tested with a small knife. From time to time, shake the pot without removing the lid to prevent the bottom potatoes from burning. Turn off the heat and allow the potatoes to steam for another 5 minutes. Don't overcook. Toss with the dill, and serve hot.