If you do what JBean is suggesting, be sure to UNSHARE every drive you shared in the file transfer process BEFORE you connect the PC to a broadband ISP. Don\'t share drives while it\'s connected. You could be creating a major security hole if you do that.
I think you should just swap the old hard drive to the new PC. In fact, you don\'t even need to copy the files to the new drive. Just leave it permanently connected to the new PC as a slave drive, and pull off the files whenever you need them.