For everybody else but that one guy, you can read this too
Anyway, I built a very simple frame, pictured below. I used the copper pipes (hollow) because they don't stretch so it's easier to get the basket where it needs to go. I used the wire at the end because it is flexible and allows for adjustments to get the basket into the distilled water. You can counterweigh the frame at the top or at the base (I use a book at the base)

Then you need a decent scale and a light cup with distilled water. You need to adjust the height so that the basket is submerged in the water and not touching the bottom of the cup or the sides.

Now you just add the coin and measure the wet weight.

The basket is just a light aluminum washing machine part from the hardware store. Once you have the wet weight you can ccalculate SG = Dry weight / wet weight
I take three measurements of each for every coin. The average must be within .01 grams of each measurement other wise I take two more, drop the high and low and repeat until I get decent accuracy.