How can I copy data from an external HDD that is about to fail? I have an external HDD that is about to fail. I want to copy as much as I can to a new external HDD before it completely dies. I've been able to start copying files for a few minutes before the drive starts turning off and on repeatedly and clicking.

Is there a (preferably free) Mac application that can copy files off an about-to-fail hard drive, handle the drive suddenly disappearing, and continue where it left off when the drive is reconnected?

Or do you have any other tips that might help? I've heard rumours about putting a drive in the freezer. Do you think that might help, or is it more likely to kill the drive?
I would try using rsync from the command line. 

rsync -av --ignore-errors /Volumes/failingDrive/ /Volumes/brandNewDrive


should do the trick. Mind the trailing / at the end of the source. Rsync will not copy files it finds on the destination, so if you call it a second time it will continue where it left off.