Invisible Dice

There is a regular six-sided die with faces $1$ to $6$, and a special invisible six-sided die. You don't know what's written on the invisible die's faces: each face shows some integer, and faces may repeat. Both dice are fair and are rolled together, and I tell you only the sum of the two outcomes. It so happens that this sum is each of the integers $1, 2, \ldots, 12$ (both $1$ and $12$ included) with equal probability $\frac{1}{12}$. Is it possible to label the invisible die so that this happens?