Some heavy gauge copper wiring (or sheet copper in my case), a starter solenoid, a microswitch, and two copper contacts (solid copper roofing nails).
I added a bit of polycarbonate and some rivets to keep everything in place, and soldered the nails in using a micro torch (not in that order).
The 150-200A of current delivered by the battery is just about right for the short, high current pulses needed to make a cell welder. You physically provide the short pulse width, with a tap to the microswitch.
Derived from this video I saw on YouTube. Dead simple.