This is pretty much just what happens to bettas when they aren't housed properly. You should return the 1 gallon bowl if you still can because it is pretty much as bad as the half-gallon. How long have you had him for? Housing your betta in anything smaller than 2 gallons (despite the prevalence of unacceptably small betta homes) presents multiple problems:
1) Bettas are tropical fish and NEED a heater to survive properly. There are currently no heaters on the market that are not too powerful to heat anything less than 2 gallons safely. Believe me, I have searched long and hard. Lower than preferable temperatures will weaken your fish's immune system and make it particularly susceptible to a disease called 'Ich'. You'll want to shoot for a temperature of around 78-82, with 80 being preferable.
2) All fish produce a toxic substance called ammonia in their waste. In order to keep ammonia levels safe in a tank of 1g or smaller, you will need to change the water pretty much daily (which is WAY too stressful for any fish to survive in the long term). If not, your betta will be 10 times more vulnerable to fin rot, ammonia poisoning, fungus, and a slough of other problems.
3) Contrary to popular belief, bettas need just as much space as any other fish. See the 'Mud Puddles Mythbuster' pinned in the housing section for details. The stress your betta undergoes from being kept in such a tight space will also hurt its immune system.
You would be surprised at how well some bettas recover from seemingly terminal illnesses. If you invest in a heated, spacious tank for your betta, I am confident he will make a full rebound. Also, keep reading betta forums to get an idea of how to care for your fish properly.
Here are some awesome links: