You have your Betta in a 1 Gallon and you only do a water change once a week? I personally find that shocking!!
I don`t mean to be rude and I know we all have to start somewhere but a weekly water change in a 1G really isn`t good enough.
*waits for Nataku*.....
Seriously ifish, ideally you should be changing your Bettas water everyday or at least every other day if you intend to continue with your 1G. A 2G tank would be better but you should still be prepared to change the water or at least a partial water change every couple of days. I doubt any of us would like to swim around in a confined space when we`d peed and pooped in the water....why should it be any different for our fish?
Am I right in presuming you don`t have a heater either?