A water bath canner can't safely preserve potatoes or other low-acid vegetables, according to the National Center for Home Food Preservation. Use a pressure canner instead, in order to minimize the risk of botulism.
Low-acid foods are at risk of contamination from Clostridium botulinum, a dangerous pathogen that can survive boiling. Only a sustained temperature of 240 degrees Fahrenheit can kill it, and this requires a pressure canner.
Boil 1/2-inch potato cubes for two minutes, and then drain them. Whole potatoes require 10 minutes of boiling. Pack the hot potatoes loosely into sterilized canning jars, stopping 1 inch from the top. Top up the jars with boiling water, making sure the potatoes are completely immersed. Cover the jars with sterile lids, and screw the rings finger-tight.
Check the canner's seal and valves as directed in the manufacturer's documentation. Bring 2 to 3 inches of water to a low boil in the canner, and then load it with jars. Open the pressure valve, and lock the lid into place. Vent steam from the canner for 10 minutes, close the vent and bring it up to pressure.
Processing time and the pressure required vary with altitude. At sea level, pint jars of potatoes require 35 minutes at a pressure of 11 pounds per square inch. Quart jars need 40 minutes. At higher altitudes, increase pressure as directed by the canner's manufacturer or as indicated by the NCHFP.
Turn off the heat at the end of this time, and let it rest until the pressure releases naturally. Remove the jars, and allow them to cool to room temperature. Any jars that have not sealed properly should be refrigerated and used immediately.