What's the best way to install a new throttle cable?

I got some awesome tips on diagnosing my moped's stuck throttle cable problem. The solution seems to be to install a new throttle cable.

Therein lies yet another problem. The throttle cable routes through the frame of the moped. What's the best way to route the new throttle cable through the frame without taking the frame apart (if taking it apart is even possible)? Once I get the cable through the frame, how do I connect it to the actual throttle on the moped?

For reference to the original problem, take a look at my throttle cable teardown.

Thanks a lot!