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I'm beginning my reasoning this way: one child in a black hat looks at the other children and thinks, 'If I had a white hat, then each of the black-hat kids that I see would have seen 4 kids in white hats and 1 kid in a black hat...'

So. The kid in the black hat is observing two other black-hat kids and thinks, 'If I had a white hat, these two would have stared at each other, each wondering if the black hat that they see is the necessary black hat from the puzzle and if it is the only one.
But they are not staring at one another but keep glancing at me as well. Which must mean that I am wearing a black hat, too'.

I agree that some children would be able to solve this puzzle with considerable accuracy by using this method - It just depends on their ability to read body language. However, it might fail if the children were aware of what you were doing, and decided to somehow try to deceive you. The other problem is that the children's first instinct would be to look at each others hats; not each others eyes. By the time any of the children figured out that they could gain useful information by watching the other children's eyes, they would have missed all the "action". Still, even if it was fool-proof, that's not the answer I had in mind and so the puzzle remains unsolved.

So... Is that the answer? If people are looking at you you can guess that you have a black hat?

Thinking out loud here: each kid looks out and sees a 3/2 ratio of hat colors. They can speculate from this that, either, the ratio including theirs is 4/2 or 3/3. They are told that there is at least one white and one black, which is immediately observable when seeing the other kids, and so, seemingly moot. Having no other information on which to go, I really don't see how any of the kids can figure it out.

If each kid sees 2 hats of one color and 3 of another, they can assume they are most likely wearing a hat of the first color. Unless they are able to communicate what color hats they see, there's no way to truly "solve" the puzzle, only make an educated guess on the color of their hat.

Well you say they MUST be told that there is at least one black hat, but even if they where not told they will come to the conclusion there are in fact at least two of each. So because that little detail is added it makes me think there is more to it by the fact it was told, because they would know it even if they where not told.

The only reason they'd have to be told about the colors is if they were blind, in which case they couldn't solve it.

Let's suppose that the children are not greedy and the spirit of working as a team is more important to them than that one ice-cream they've been promised. Then the scenario could be as follows:
One ingenious child - say in a black hat - takes the initiative of taking the other children by the hands and grouping them according to their hat colour. Thus, two groups: 2 children in black hats and 3 children in white hats - are formed. Then the teamwork spirit comes into play: the children in black hats realize what their fellow is doing and drag him/her into their group.
So the first child realizes what the colour of his/her hat is - thanks to the cooperation with the other children.
But then again, the one-black-hat-one-whiter-hat is rendered useless.
Which makes this just ruminations on the topic and not the correct answer.

When I originally posted the puzzle, I said that the children were told that there was at least one black hat. I later changed it so that the children were told that there was at least one white hat as well. If I hadn't made that change, it would only be solvable by those wearing black hats. Now, it can be solved by those with white hats as well. Of course, the method for solving the puzzle remained the same.

A child could mentally calculate the probability of his/her hat being, say, black, using the Bernoulli trial or some other probability theory formula.

In the original question, I said that, "Being very intelligent children, they all spent some time considering the puzzle before simply guessing." By that, I mean that none of the children are going to just guess until they're absolutely certain that the puzzle can't be solved. So the children have all the time in the world to come up with an answer. Would it change anything if, say, an hour had passed?

The one-hat condition could make them stand and think for a pretty long time, just as it has made us. In turn, if they stand and think that long, and it's a hot sunny day (after all, the reward IS an ice-cream ), the children in black hats would start feeling uncomfortable because of the heat, since black colour absorbs heat. So the reasoning of each of them would be, 'If I was wearing a white hat, I wouldn't feel so hot. So I must be wearing a black hat'.

if they can consider at an infinite amount of time, eventually one will get thirsty, ask for a glass of water, and use the glass as a mirror!

Each child sees three hats of one colour, and two of the other, so they are trying to decide whether the ratio is 3:3 or 2:4. Any child could solve the puzzle by considering what would happen if the ratio was 2:4. With two black hats and four white hats, the children wearing the two black hats would be able to solve the puzzle with relative ease. Observing that they are unable to do so essentially guarantees that the ratio is 3:3.

I suspect that most of us understand how the puzzle could be solved with a 2:4 ratio, but if not, here is is. Let's say that there are two black hats and four white ones. The children wearing one of the two black hats would see one black hat and four white hats. Considering that if their own hat was white, the child wearing the only black hat would solve the puzzle instantly, observing that they seem unable to do so would enable them to deduce that their own hat was also black.

Two: The North and South Poles.