What glue should I use for permanent LEGO construction? The incredible precision of LEGO parts makes them attractive for building precise jigs, but the fact that they can come apart fairly easily isn't so helpful for this application. Can anyone recommend methods and materials for gluing LEGO bricks together to create permanent assemblies?

I'd prefer to avoid thick adhesives like epoxy or hot melt glue, as I suspect that those might make it difficult to get the bricks to snap together completely. I think most bricks are made of ABS plastic, so I'd guess that a solvent like acetone might work to weld them together, but I've never tried it. Does anyone have experience gluing bricks together?
The LEGO Group have recently started using solvents to "weld" elements together (notably the Minifigure legs to the magnet blocks), which are fairly heavily controlled in pure forms in most countries under drug legislation.


  I have discovered that LEGO is not using glue at all, but a solvent, that actually loosens the surface of the ABS plastic and allows the molecules to combine. Once the solvent evaporates, it leaves behind a cleanly fused surface with no residue at all.
  
  Reading further on the subject, I found that LEGO has been using two types of solvents. One is called MEK (methyl-ethyl-ketone) and at some point they switched to GBL (gamma-butyrolactone). I’m no chemist, so I had no idea what these are, but I read that both of them are water soluble at certain temperatures. So, I thought to try loosening the seal between the minifig and the magnet by hot water.


However, as the opaque LEGO bricks are made from a styrene based plastic (ABS), most "Plastic Cement" or "Polystyrene Cement" glues should work to fix your parts together permanently (as these typically contain the same solvents as TLG use, i.e. MEK/Butanone). As with most other plastic gluing projects, care should be taken around transparent elements, as well as ensuring you don't use too much and have it leak over the exterior of your model.