Air Conditioning - Compressor will not turn off

Over the weekend I realized that my A/C was running but no air was flowing into the house. I removed the panels and realized that the system was frozen over. After doing the defrost and cleaning, I put everything back together, and returned power to the unit. The moment I apply power the compressor fan will start running and will not stop running. I have cleaned out the compartment where the wireing is (since about 2 years ago I had a tech come out because ants had gotten in and caused a short and caused the same problem). I have replaced the thermostat with a new digital one (since I needed to upgrade anyways. Anyone have any suggestion on what else to check?

If I understand correctly you're saying that the outdoor unit runs even if the thermostat is off and not for calling for a cooling cycle? The most common culprit is the compressor contactor which is a large magnetic relay in the outdoor unit that operates the compressor and fan motors. What usually happens is the contacts become pitted/dirty and they weld together in the closed position. Replacement is within DIY skill levels depending on how much you know about electricity and how comfortable you are working around it. To replace contactor, shut off power to both the indoor and outdoor units, make a sketch of where the wires go or take some photos. Buy a new contactor that has at least the same ratings as the old one, install and rewire it. Check your work before powering up. The voltage for the magnetic coil is 24 volts. The contact points will have an 2 amperage ratings, one is in resistive amps and the second is inductive amps[the resistive amps rating is larger than the inductive amps rating, the one to pay attention to for this application is the inductive amps rating]. Purchase a new contactor that has at least the same inductive amps rating as the old one; it's ok to go higher but not lower. For example, if your contactor is rated for 25 amps inductive, you could replace it with one rated for 30 amps but would not want to use one rated for 20 amps.