Have to agree with Styg.. it's most likely due to lack of water conditioner and heat.
Water conditioner info
When acclimating fish to new water it's best to not only float it to get the temp the same, but to get them acclimated to the new chemistry- by removing part of the water from the cup (throw it out) and replacing it with water from the tank a couple of time in the 20-30 minutes he is floating.
You will want to at least get a thermometer to check the temps to make sure you aren't refilling it with too cold of water, or too warm of water. Your idea temps would be 76-82*F. Even if you are floating him in to even out the temp, doesn't mean the temp you are placing him into is safe.
I would ask someone for the $2 needed (it's less then that at Walmart) for a thermometer at the least.
In a 1 gallon unfiltered you will want to do 1 50% and 1 100% water change per week- using water conditioner at full dose for that size each time. It's very important to make sure you have all the appropriate supplies needed when taking care of an animal- and even though they seem "easy" bettas (and other fish) are anything but when you are first learning. But once you have all the correct supplies it does get easier.
If you have any questions, please feel free to ask!