I think you're looking for <a href="http://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)">Liouville's theorem</a>.  This theorem states that for $n >2$, if $V_1,V_2 \subset \mathbb{R}^n$ are open subsets and $f : V_1 \rightarrow V_2$ is a smooth conformal map, then $f$ is the restriction of a higher-dimensional analogue of a Mobius transformation.

By the way, observe that there are no assumptions on the topology of the $V_i$ -- they don't have to be simply-connected, etc.

----
EDIT : I'm updating this ancient answer to 
<a href="http://lamington.wordpress.com/2013/10/28/liouville-illiouminated/">link</a> to a blog post by Danny Calegari which contains a sketch of a beautifully geometric argument for Liouville's theorem.