"General Tech" is exactly the right place for this question.
You can't solder exhaust, but if you can weld then that's the preferred repair. Else you can replace parts of it with cheap universal exhaust parts and clamps allow you to install without welding. If done right it's an acceptable repair but probably only lasts a couple years.
So, the rear of the muffler, where it comes out and into the tail pipe (or directly into an exhaust tip) is rotten. Is the rot in the attached pipe, or in the muffler's own outlet?
For reference, here's a picture showing the muffler's outlet. If that part is rotted out, you will probably need to replace the muffler before it will pass any kind of inspection.
![](http://static.howstuffworks.com/gif/muffler-cutopen.jpg)
There's a slight possibility that you could get away with exhaust "bandage" type products.
Luckily, universal mufflers and exhaust pipes are very cheap.