Distributed vertex coloring is one of the classic problems and probably also the most widely studied problems in the area of distributed graph algorithms. We present a new randomized distributed vertex coloring algorithm for the standard CONGEST model, where the network is modeled as an $n$-node graph $G$, and where the nodes of $G$ operate in synchronous communication rounds in which they can exchange $O(\log n)$-bit messages over all the edges of $G$. For graphs with maximum degree $Δ$, we show that the $(Δ+1)$-list coloring problem (and therefore also the standard $(Δ+1)$-coloring problem) can be solved in $O(\log^5\log n)$ rounds. Previously such a result was only known for the significantly more powerful LOCAL model, where in each round, neighboring nodes can exchange messages of arbitrary size. The best previous $(Δ+1)$-coloring algorithm in the CONGEST model had a running time of $O(\logΔ+ \log^6\log n)$ rounds. As a function of $n$ alone, the best previous algorithm therefore had a round complexity of $O(\log n)$, which is a bound that can also be achieved by a naïve folklore algorithm. For large maximum degree $Δ$, our algorithm hence is an exponential improvement over the previous state of the art.