Fin Rot Contagious?

It's caused by bacteria so yes it is. This doesn't necessarily mean the other fish will get it, but it should be a wake-up call to you to up your maintenance.
 
Fin rot is always caused by poor water conditions. The one fish might just have a better immune system at the moment. Check your water (ammonia, nitrite, nitrate, pH) and begin changing it a lot more often. If you can provide pristine water conditions, I have had fin rot heal on its own just with lots of water changes. Good luck!
 
One other thing not mentioned is temperature...make sure your betta's temperature is staying above 74, that'll help get those immune systems going again.
 
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