If I understand you correctly, this is what I do..
1. Select the curve (edge) that you want to extract (duplicate) .
2. Go into the "Surfaces" Menu Set (i have version 2009) and under the "Curves" Menu select "Duplicate Surface Curves"
3. Immediately select "Attach Curves" under the "Edit Curves" menu and then with the curve still selected, delete history. Deleting history will unlink the attached curve from the "in pieces" curve because at this point you will have 2 curves on top of each other. One will be a complete curve. The other will be in pieces. Just delete the "in pieces" curve.
Note: These 2 curves are in addition to the original curve that is still apart of the geometry.
Don't be satisfied with what you can do but rather strive to do the things you can't do!