I went to the plumbing section at Walmart. I picked up this soldering torch for $15. In the same isle they sell lead free silver solder. The solder I bought was $10 and also came with a container of flux, an application brush, and a piece of sand paper. So total damage: $25. These earrings will only use a small fraction of that but you’ll get to use it for all kinds of other projects.
Basically I understand it this way. Solder is a low melting alloy. Flux is a paste that helps solder stick to metal. A torch can heat metal hot enough to the point it will melt the solder. So fire makes heat, flux helps the bond, and solder is the bond.
So this is what I did.
Step 1: Cut Off the Threads
Place the screw head on a Phillips head screw driver. File away the remaining threads until the back side is flush. I used a grinding attachment in my drill press.
If the heads are shallow enough you should end up with a perforation in the back of the head.
Step 2: Set Up for Soldering
You only need a small amount of solder. Hammer a piece of solder and cut a square out with diagonal cutting pliers.
Place the assembly in the jaws of some pliers. Position the needle so it's perpendicular to the screw head.
Place the square of solder on top.
Step 3: Solder
Step 4: Remove the Excess Solder
Polish the heads with 2000 grit sand paper, polishing compound, and a polishing wheel.
Step 5: Prepare the Backings
File the ends of the backings until no sharp edges are left behind.
Borrow some backings from an old pair of earrings and place them on your new earrings.