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Edward, We recommend waiting
Edward, We recommend waiting until at least the flower die and the vines are tending to. Then, pick a couple of potatoes and check them: "New" potatoes—the first ones of the season typically about 1.5 inches in diameter, should be used immediately. They do not store well. Leave a portion to harvest when all of the vines have died back (to a point in your question, the key word there is "died"). "Cure" (thicken the skin of) those before putting them away for storage by putting them in a space with temp of 50°F to 60°F and relatively high humidity (85 to 90 percent), such as a basement, for two weeks. Remember, too,not to store potatoes with apples or other fruit. Ripening fruit gives off ethylene gas, which encourages the tubers to sprout. Wishing you a hefty harvest!