thermodynamics - How much more energy does it take for a human body to heat 0C ice vs 0C water?

I'm trying to determine if going through the trouble of ingesting ice is worth the hassle versus ingesting ice-cold water, but my physics skills are rusty.

If I drink a gram of ice water at ~0C, my body has to heat the water to 37C.

Per wikipedia, the water will be heated by

$$1\text{ g} \times \underbrace{37\text{ K}}_\text{Temperature difference} \times \underbrace{ 4.1813 \frac{\text{J}}{\text{g K}}}_\text{Specific heat of water} = 155\text{ J}$$

Whereas if I ingest a gram of ice, in addition to those 155 Joules, it will need to heat by 334 Joules to do the transformation from ice to liquid. So the payoff to ingesting ice instead of water is about 3 times more calories burned.

Is my reasoning sound?

Note: You could turn my question into a meta-question by attacking the logic of ingesting cold water for burning calories, but please be so kind as to keep your answers to the physics question I've stated.

Answer

I think your reasoning is sound.

However, being in Melbourne, Australia (which is heading for 35 C today), I can think of better applications of eating ice than burning calories. You would certainly experience a greater drop in body temperature by consuming equal amounts of ice versus water at 0 C.