A little more info would be helpful: http://www.bettafish.com/showthread.php?t=49233
I want to make sure we don't miss anything. 2 gallon is minimum for keeping bettas btw and even that will require 2 weekly water changes.
A half gallon requires daily water changes alternating between 50% and 100%. 1 gallon should probably be getting 3 weekly changes but some will do two.
In the future, watch your betta eat and make sure no pellets fall to the bottom. If they do, fish them out. I find a turkey baster dedicated just to him (so no soap or chems every touched it) works best.
It's very important at this point to keep his water very clean. If you're keeping him in the 1 gallon do a water change every day of at least 50% and 100% alternating days and the 100% should include a thorough rinse of all gravel and ornaments. To treat minor fin rot like that, I would use aquarium salt, predissolved at 1 tsp per gallon. You can do this for up to 10 days or until you see the fins stop dissolving and you actually see a little regrowth. Aquarium salt should be redosed along with however much water you remove.. so if you remove 50% or 1/2 gallon of water add 1/2 tsp back. If you do a full change add the full 1 tsp back. Then you know you're in the clear. Stress Coat at 1 ml per 1 gallon is also helpful at regrowing fins. If this doesn't work, and it really should, you can try some antibiotics.