Switch off the home's main breaker so there is no power running to the air conditioner while you are working.
Remove the cover of the thermostat. Loosen the screws that hold the thermostat onto the mounting plate. Carefully take the thermostat off of the mounting plate without damaging the glass mercury tube inside.
Mark the location of each thermostat wire with masking tape. Remove the terminal screws and disconnect the wires from the thermostat. Loosen the screws holding the mounting plate on the wall and remove the plate. Inspect the wire ends for any signs of fraying or splitting. Cut any frayed ends off and strip a new end with wire cutters. Wrap the loose ends around a pencil or stick to keep them from falling back inside the wall.
Hold the new mounting plate up on the wall and mark the position with a pencil. Use a level to make sure the line is straight. If the thermostat is not level, the liquid mercury will give inaccurate readings. Mark the location of the screw holes when you are satisfied with the location.
Use an electric drill to make holes where you have marked the screws. Insert plastic screw anchors into the holes, pressing them down as far as possible so they do not stick out of the wall.
Unwrap the loose wires from the pencil and thread them through the holes in the new mounting plate. Attach the mounting plate to the wall and tighten the screws into the plastic anchors. Re-check the mounting plate to be sure it is still level.
Attach the wires to the new thermostat and tighten the terminal screws. Place the thermostat on the mounting plate. Check the thermostat's user guide for instructions on how to set the clock and program the air conditioner's schedule. Attach the front cover and turn the home's breaker switch back on.