I could be wrong about this, but I think that neither Nick's robocopy solution or Nathan's copy command solution will work. Robocopy seems to want a mapped drive, which I can't do because it's not running on port 80 or 443, and Windows can't handle a path that includes the port in a UNC-style name:
copy src \\dest-machine:45000\shared-library-name\dest
I didn't see a NAnt task that would handle this situation, either.
I also know there are tools that will allow me to map WebDAV drives. I was more wondering if there was some way to make use of the Network Places that have been set up without using the GUI.