1 cubic inch is 2.54 * 2.54 * 2.54 = 16.4 cc = 16.4 grams. 1 F = 5/9 C. The heat capacity of water is 1 cal/gm.C. So the number of calories is 16.4 * 5/9= 9.1
9.1 cal / F / cubic inch * ( 98 F - 34 F) = 582 cal / cubic inch
Edit: Yes, I realize 1 Cal = 1000 cal, I just wanted to clarify the math.
1ml of water weights 1 gram, and raising this ml of water one degree celsius is 1 (little c) calorie.
Drinking 1 ml of ice water (at about 0 degrees celsius) and raising it to body temperature (at about 37 degrees celsius) thus takes about 37 (little c) calories.
Thus drinking a liter of ice cold water would consume about 37 (big C) Calories.
One could drink 15L per day (about one liter per hour and 5x the recommended minimum intake) and burn about one pound of fat per week.
A grand total of 1.8kcal per ice cube. Only 1000 ice cubes and you could eat twice as much as usual!
At around 16 liters of water once melted that would be an interesting Darwin Award for sure.
Minor nitpick but: Typical glass of water weights 250g, but typical (even obese) human weight is very far from 250kg :) Guess you put one too many zero.