Finding a small dominating set is one of the most fundamental problems of traditional graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary parameter $k$, our algorithm computes a dominating set of expected size $\bigO{k\Delta^{2/k}\log\Delta|\dsopt|}$ in $\bigO{k^2}$ rounds where each node has to send $\bigO{k^2\Delta}$ messages of size $\bigO{\log\Delta}$. This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of rounds.