Woahhhh!!! That temperature is more than likely why your fishy is at the bottom of the tank. He's probably feeling quite lethargic from the cold. You say that you have a heater in your tank, but it isn't going above 68*F? It might be faulty/busted. :( It should stay at about 79*F (I only say this, because in my experience, even my non-adjustables have never gone below 75*F). I would go buy another submersible heater for you fish, if I were you. Until then, try putting a towel around the tank. I would put one in the dryer for a little bit to heat it up, then wrap the tank in it. Also, if you can get the tank to a higher spot, then that will help as well. The reason is that bettas are tropical fish and like warm water. 80*F is usually agreed to be the perfect temperature, but they can survive anywhere from 78*F(minimum)-86*F(maximum).
How often do you clean your tank and how many pellets do you feed him a day?