Your engagement ring is probably one of those things you can't risk losing. While I prefer having the ring in hand to work from, I understand not wanting to mail it in. Instead I have made some instructions on how to trace the curve of your band, so all you need to mail me is a strip of paper.
Here's a link to order a custom curved band.
First, find some thicker paper, cardstock works well. Then cut out a 1 inch by 5 inch strip.
Now take the strip and give it a little curve by wrapping it around something like a chunky marker. Don't curve it up to tight, just adding curl to the paper.
Then, with your paper tube gently snug it inside of your engagement ring. Try keeping it a tight fit. Also put the paper end on the bottom of the band, not under the curve area.
Now hold the paper and ring to keep them from moving, and trace the outside of the band. Hold the paper still while turning and get both sides.
Your strip should look something like this. For good measure, do this one more time and send me both strips so I can get the best tracing. (For reference I also prefer one or two photos of the engagement ring.)
Custom Band with Ring