Solder to it or use a crimp on connector designed to work with that type of wire. For example, crimp on connectors are often used for TV cable coax, a type of shielded wire. A solder joint to the foil is pretty weak, so be sure to leave some slack or use some sort of strain relief.
Right, I was thinking the same thing regarding the crimp on connector when I mentioned the idea of soldering a female RCA pigtail to both parts and running a standard RCA audio interconnect cable (which is the same idea as TV cable coax) between them. The connectors for coax, whether they are RCA or F-Type, connect to the center conductor of the coax cable on their center pin and the outer rim connects to the outer conductor (braided wire), which in turn is contacting the foil shielding. In fact, doesn't the braided wire itself contribute to the shielding, in addition to the foil?
So I think the best way to do it would be to get a couple of female RCA pigtails, which is just a female RCA connector with two ordinary wires attached, one connected to the center pin and the other connected to the outer rim. With one of these pigtails, I'll connect the outer rim wire to ground on the ground pinout of the game board, and the other I'll connect to positive ("sound"). With the second pigtail, I'll connect it to the ground and "sound" points on the amplifier in the same fashion. Then I can take any RCA cable that I have kicking around and connect between the two pigtails. This is exactly how any other audio setup works to get the low level signal from the preamp to the amp, so it ought to be good here as well. Then from the amp to the speakers, you just run ordinary "speaker wire".
Does this sound about right to everyone?