The tank doesn't look filtered and this is your problem. All fish produce ammonia (poop as you're American!) as does decomposing food (you won't get all uneaten food out of the tank). Ammonia kills fish.
First - go change all the water NOW. Add the water treatment etc, change all the water then read the rest of this post.
A filter isn't really there to take bits and pieces out of the water (mechanical filtration) although it does this to an extent. What we're really interested in with fish keeping is the biological filtration. On the filter grow special bacteria that eat ammonia and turn this into nitrIte. Problem is nitrIte also kills fish (quicker than ammonia sometimes).
But, over time more bacteria grow which eat nitrIte (ain't nature cool) and turns this into nitrAte which doesn't kill fish in small quantities and is taken care of by water changes.
Basically, without a filter your fish is being poisoned to death and will die very soon without radical intervention. in a 1gal tank you need to be changing all the water every day to even stand a chance of it surviving. Once a week is not nearly enough - in a small tank there is less water to dilute the ammonia so the problem is even more severe
You then need to get a filter and whilst you're at it a 5gal tank minimum.
I know that these Betta bowls etc are sold and they just shouldn't be. I'm not blaming you at all for what's happened - unfortunately shops and websites that sell such aquarium equipment can give out the most appalling advice.
So to start with change ALL the water NOW and then EVERY day or as its only 1gal maybe twice a day.
Then, get a bigger tank and a filter and read up on cycling the tank - there's loads of information on here. You can't fishless cycle as you already have the Betta but with a small tank big water changes aren't so hard to do.
The reason the fish has started going down hill is because it takes time for the ammonia to start poisoning the fish. If you act now you may well save him.
Hope this helps and please ask if you are unsure on anything I've put.
Cheers
Danny B