Yes, it can, but you'd have to have your API test kit and test the water daily, and be ontop of the water changes so that your betta won't get sick from the ammonia. Because my little nutshell on cycling is rather brief and doesn't go into details, you'll want to look at this site for more info.
I'll give you fish- in cycling in a nut shell:
-a tank full of water
-a filter with some sort of sponge in it (no carbon cartridges necessary)
-a source of ammonia (in this case I guess it'll have to be your fish since you have nowhere to put him in the meantime)
-a water test kit (I HIGHLY recommend the API Master Test Kit- http://reviews.walmart.com/1336/3635493/freshwater-master-test-kit-reviews/reviews.htm
1) set up tank and introduce your constant source of ammonia (your fish)
2) Do daily water checks with your tester on the ammonia, nitrItes, and nitrAtes. It is a good idea to log them so you can watch the patterns.
3)If you see the ammonia or nitrItes get over 0.25 ppms of ammonia or nitrItes, do a 50% water change this may be daily (or not depending on how many plants yo have), but chances are it'll be around once a week.
4) continue this until you notice that the ammonia is 0, the nitrItes are 0, and you have had nitrAtes 3-10 ppm for several days without water changes.
5) you're done
* Important: 1) Once you have done, you have to do once weekly water changes, the ammount of water depends on your tank size, ammount of fish, and ammounts of life plants. 2) never ever ever scrub down your tank, because that will be scrubbing the good beficial bacteria off. 3) Filter media is important. If you have a filter sponge in there, never remove it. When it gets dirty, rinse in a bucket of tank water. 4) The carbon pads you buy aren't necessary.