My wife and I were cookin' a Cajun feast for the anniversaire de ma mere. While I handled the vittles, the lovely and talented Mrs. Boy made the drinks.
  She had made the tea strong and wanted to dilute it with 4 cups of water but the guests were at the door and the tea was still hot so she decided to dilute it with ice instead.
  She turned to me and said, "Fat, sweetie, how many ice cubes make a cup of water?"
  I confessed that I did not know as I had not measured the water when I made the cubes. To make matters worse I had not paid attention to how full I had made the trays so we couldn't just refill them and see how much they held.
  Things seemed desperate, as I'd die before I'd serve my Gumbo without sweet tea, but Mrs. Boy is no fool and she found a way. The tea was just right (though the cheese grits were a little burnt).

How did Mrs. B manage to ascertain the proper number of ice cubes to produce the 4 cups of water needed to dilute the tea? All she had to use was the ice cubes themselves, an ungraduated glass pitcher of unknown volume and the 4 cup graduated Pyrex measuring cup full of (too strong) tea.

I believe you're right when you say that the ice cubes must be equivalent in weight to four cups of water, but I don't think your method will work.

So if one ice cube displaced a cup of water, then one ice cube was equivalent in weight to one cup of water

I think that an ice cube that displaced a cup of water would be equivalent in volume to a cup of water. If you've ever left a bottle of water in the freezer you might remember that water expands when it freezes, however. I believe Archimedes method would tell you the volume of ice, but it would not give you exactly four cups of water when it melted, because the volume occupied by it would decrease.