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:

However:
CompleteGraph[4,
EdgeWeight -> Range@6,
VertexShapeFunction -> "Name",
EdgeLabels -> "EdgeWeight"]
Results in:

So, no Edge Labels … I guess it is a bug.
I used a nasty construct like:
adj = {{\[Infinity], 1, 1, 1, 1}, {1, \[Infinity], 2, 2, 2},
{1, 2, \[Infinity], 2, 2}, {1, 2, 2, \[Infinity], 2},
{1, 2, 2, 2, \[Infinity]}};
WeightedAdjacencyGraph[adj,
VertexShapeFunction -> "Name",
EdgeLabels ->
MapThread[Rule,{EdgeList@#,AbsoluteOptions[#, EdgeWeight]/.{_ -> x_}-> x}],
GraphHighlight -> FindEdgeCover[#]]
&@ WeightedAdjacencyGraph[adj]

Better ideas?
For a regular
GraphPlot, you will need a slightly more complicated solution usingEdgeRenderingFunction(documentation). Suppose you have an adjacency matrix where the elements are also the (directional) weights.Here are some labels for the vertices, supposing you are drawing network diagrams for international inter-bank exposures (the original has a lot more countries!).
The following does what you need. The tricks are the reference back to the adjacency matrix using the parts of
#2inside the part specification, to reference the correct elements ofnums, and theMean[#1]to locate the label at the midpoint of the edge. The slot#1seems to hold the coordinates of the vertices.