This shows my Brompton folding bicycle, which I converted from chain to belt drive. I'm very happy with it. It's very quiet compared to a chain, and no more oil, grease, and dirt!
The images show the setup. The frame had to be cut to allow the belt to be fitted. Those little metal plates and small bolts hold the frame together (the frame is cut right in the center of the dropout).
The "CenterTrack" belt is from Gates, as is the "chainring" and "sprocket" (roughly $80 each, carbondrivesystems.com). The chainring has a standard bolt-circle diameter, which required the Brompton cranks to be replaced. (It turns out BMX bikes are the main use for single-ring cranks, which accounts for the color choices.) The sprocket I got is for a Shimano Alfine hub, but it's roughly compatible with the Sturmey Archer five-speed hub on my bike – the Shimano has nine splines, the Sturmey Archer only three. An hour or so of filing...
The idler sprocket, which has the primary purpose of taking up the slack in the belt when the bike is folded, is plastic - I made it with 3D printing (shapeways.com). While it took quite some time to learn how to specify the part (with the Openscad computer-aided-design program), actually having it "printed" cost less than $12!
The standard chain-drive Brompton has a second idler that helps "fold" the chain. I replaced the second idler with a little "hook" (you need something – otherwise there's too much slack belt in the folded position). I tried a second small idler, but the belt is stiffer than a chain, and didn't want to run around the idlers in any case. As a result, I'm not able to reposition the pedals once the bike is folded. I've learned to position the pedals correctly before folding the bike.
I worry a bit about the tight bend the belt makes around the hook in the folded position – you're not supposed to kink the belts – but so far so good.
As long as one has an internal-geared hub, I don't see any point to a chain – the belt seems to have all the advantage!