I'm sorry, but absolutely NO fish should be kept in a 1 gallon.
The ABSOLUTE MINIMUM for a betta is 2.5 gallons, though they do much better in 5 gallons. They also NEED both a heater and a filter! All fish need filters, and the betta needs a heater because they are a tropical fish, and tropical fish need warm water to live happily and healthily!
I would rethink owning your betta, unless you can provide him with a sufficient home.