Can a betta live in a 1 gallon tank?

Yes, but this is the smallest recommended size for a betta. Larger tanks are recommended and some people will tell you that 2.5 or even 5 gallons is the minimum. I agree, but personally I think that it's better that the fish be kept in a one gallon tank by somebody who cares about its welfare, than left to rot in a cup by somebody who treats it like an ornament. If keeping a betta in a one gallon tank with no filtration, you will need to change half the water every second or third day to stop its waste from building up to harmful levels. Once a fortnight, you should put the betta in a large, clean cup and rinse the gravel and decorations under the tap to clean them thoroughly. Don't forget to use chlorine neutraliser whenever you change the water. If you do not overfeed your betta, this should be suitable to maintain good water quality. If he develops finrot or is always lethargic, this is a sign that the water quality is not good enough and you should do water changes more frequently.