He could of died from various differea few things, however a few things stick out.
1) tank size - IMO, I feel that they should have at least 3 gallons of water. However a 1 gallon IS doable but it requires more matience - ie, water changes. In general, the larger the tank, the easier they are to maintain
2) Lack of a heater - Despite what many people seem to think, bettas need heaters, unless you live in like Thailand. If they are constintly exposed to cold water - under about 78 degreese - they become more suspectible to disease. Tank size is important for this becuase smaller tanks are harder to heat. Many heaters are too tall to fit in smaller tanks or they are not adjustable. SOme like the Hagen 25 watt are on the short side so they work well.
3) Water conditioner. I also didn't use it when i first started and my fish kept dying in less then a week. Finally I figured out they were dying from heavy metal pisioning from whatever they add to tap water. There are many good brands of water conditioner. Prime and Stress coat seem to be the most popular.
4) Fins always folded. I am also assuming that by this you mean clamped fins. DId they look all closed up - like this (pic stolen from google)
Clamped fins are usually one of the first signs that something is wrong, however that "something" does not always mean it's sick per se. It could indicate something is wrong with the water like its too cold or the ammonia is getting too high.
It's really hard to say exactly why the fish died but if you fix the problems we have pointed out - your next betta should be a happy fishie.