Pull the alternator and have it tested. I've replaced mine and don't remember it being any big deal - it's at the top.
If the problem was that the belt is too loose, then it would run with the belt on just as it does with the belt off. My suggestion to pull the belt was to see if you had a problem with a driven pulley or accessory - not a problem with the belt itself. I'm leaning heavily toward the alternator at this point. As mentioned above, it isn't just that it dies and quits charging. It can also overcharge, put out AC instead of DC, or the bearings can lock up. It ran with the belt off - that should tell you that you have a problem with something that is DRIVEN by the belt - not the belt itself.
And make sure that you have it routed correctly. There is a diagram in the owners manual.