A 1 gallon tank needs to be changed and cleaned out every other day. You must change 100% of the water and wash out the tank with hot water in order to get the feces, uneaten food, and ammonia residue out of the gravel, decorations, and the tank itself. The new water should match the temperature of the old water and it must be treated with a dechlorinator product such as Seachem Prime before adding the fish. The fish should be slowly acclimated to the water each time.
Bettas are tropical cold blooded animals that thrive in temperatures of 78-83 degrees, and cannot survive in temperatures that low for long. I highly suggest getting your fish a heater and a larger container that can be heated as soon as possible. The temperature difference may not seem dramatic to warm-blooded mammals like us, but keep in mind that the entire metabolism of cold blooded animals is dependent upon temperature, without adequate heat your fish will have digestive and circulatory problems and will have a weakened immune system, leaving him very vulnerable to illness.
For now, I suggest you move him to the warmest, most stable place in your house. If you cannot afford to upgrade to a larger tank that can be heated, I suggest going to WalMart or Target and buying a plastic rubbermaid or sterilite storage bin. These bins are only about $3 for a 4 gallon container and can be heated safely. You will also not have to clean them as often. They make great temporary tanks, hospital tanks, and quarantine tanks.
I highly recommend this heater: http://www.fosterandsmithaquatics.com/product/prod_display.cfm?pcatid=11368
the heater you buy should be low wattage and it should have an adjustable temperature dial. Non-adjustable heaters don't have a thermostat and either won't heat the water enough or will overheat the water. Don't make the mistake of getting a cheap non-adjustable heater.