We present observational data for two long-period and three dynamically new comets observed at heliocentric distances between 5.8 to 14.0 AU. All of the comets exhibited activity beyond the distance at which water ice sublimation can be significant. We have conducted experiments on gas-laden amorphous ice samples and show that considerable gas emission occurs when the ice is heated below the temperature of the amorphous-crystalline ice phase transition ( T̃137 K). We propose that annealing of amorphous water ice is the driver of activity in comets as they first enter the inner Solar System. Experimental data show that large grains can be ejected at low velocity during annealing and that the rate of brightening of the comet should decrease as the heliocentric distance decreases. These results are consistent with both historical observations of distant comet activity and with the data presented here. If observations of the onset of activity in a dynamically new comet are ever made, the distance at which this occurs would be a sensitive indicator of the temperature at which the comet had formed or represents the maximum temperature that it has experienced. New surveys such as Pan STARRS, may be able to detect these comets while they are still inactive.