Uniform Sum Distribution with Non-Fair Dice

You have two 6-sided dice, each with sides labeled 1 through 6. The dice need not be fair -- each die can have any probability distribution over its six faces, as long as the probabilities are non-negative and sum to 1. Can you choose two such dice, both non-fair, so that the sum of the two dice is uniformly distributed over $\{2, 3, \ldots, 12\}$? That is, can we have $P(\text{sum} = k) = \frac{1}{11}$ for every $k \in \{2, \ldots, 12\}$? Prove your answer.