It is possible, but not practical.Because of copper's high rate of thermal conductivity you would need a very large spotwelder. John Christiansen

I second that.

You cant afford a machine big enough to do this. It would take a huge, industrial, watercooled spotwelder, that was quite expensive.
At a minimum, you would be talking about a machine that required 100 amps of 230 volt power- about what the average small house has for its entire electrical service.

Better riveting is one answer- I dont know if you are just using copper pop rivets, but actual rivets, and a rivet set, give you a much wider range of choices of sizes, and can make better looking connections as well. Companies like JayCee http://www.rivetsinstock.com/
can sell you solid or hollow copper rivets.

You can, technically, tig weld copper as well- but again, its expensive and requires some skill. I have welded up to 1/4" copper, and it requires a hefty tig machine- 300 amps or so of power.

So here's a new thought. Untried, but I'll toss it out nonetheless. How about "spot soldering" copper?

Use a mixed flux/solder and then use a smaller spot welder to flash heat the spot to silver soldering temps. Should be doable, I think.

Rich

For building a spot welder using copper, it requires more research as well as through knowledge about spot welder.