I'm so glad I found the answer Mark had about this question.
I think it makes a lot of sense to do it your way instead of having the expense of buying other tools for such a small project.
Not to mention burning yourself if you're new at this!
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Closing Jump Rings by: Nark
I would not use a soldering gun unless I was using gold or silver rings.
So for ordinary jump rings - first you need to open them the correct way.
Using two pairs of jewelry pliers,holding on to each side of the ring - twist the right side forward and the left ring side towards yourself. Reverse to close.
Before closing, I like to add 1 small seed bead, close the ring, add a tiny drop of superglue and push the bead over the opening. Voila!
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How to Close Small Rings for Jewelry by: Christine ~how-to-make-jewelry.com
The quick answer is no, you don't need a soldering gun to close the small rings. (They are called "jump rings").