Warning! I posted the question when Mathematica v 8.0 was the coolest kid. The bug has been solved as of version 9.0.1
The help for EdgeLabels states:
EdgeLabels -> "EdgeWeight" still doesn't work in 8.0.4 and no longer seems to be in the documentation. However, here is one solution that does work:
lilnums = {{0, 2., 1., 3., 0, 6.}, {0, 0, 1., 2., 0, 0}, {1., 8., 0, 2., 0, 2.},
{10., 13., 7., 0, 0, 10.}, {0, 0, 0, 0, 0, 0}, {4., 1., 1., 2., 2., 0}}
names = {"AT", "AU", "CA", "CH", "CL", "ES"};
g = WeightedAdjacencyGraph[names, lilnums /. {0 -> \[Infinity]},
VertexShapeFunction -> "Name" , ImagePadding -> 15];
SetProperty[g, EdgeLabels -> MapThread[#1 -> #2 &,
{EdgeList[g], PropertyValue[g, EdgeWeight]}]]