What's wrong near my Home A/C thermostat? When my thermostat says the indoor temp is, say, 80, and I...

When my thermostat says the indoor temp is, say, 80, and I set the desired temp to 79, the A/C kick right in. After a short time it is much cooler surrounded by the house but the A/C never shuts off. The themostat still shows the indoor temp as 80 even though it is obviously much cooler than it have been. The A/C will run all morning until it feels like a refrigerator! I know the placing of the thermostat matter; but its placement area seems to cool dramatically beside the A/C on. I replaced the old mercury tube thermostat with an bright digital one hoping that would solve the problem, but, no change. Any ideas?
Answers:    The tstat should be placed close by the return grill on an interior wall not a exterior wall where it gets sunshine or possible a draft from down. During the winter if the furnace tends to overheat its a sign of a draft behind the Tstat. If the tstat does not fine-tuning then it maybe a impossible tstat. if the wiring is color coded then:
red is 24v
green is blower/fan
white is heat
yellow is cooling.
Rerturn the tstat and try another also turn off the ac at the tstat. if it dose not stop the contactor possibly sticking.
I really need more info on your equipment. If it is a mobile home unit or one near its own transformer than your contactor outside may be sticking. Try removing front of thermostat while its running to see if it shuts off