THIS solved my problem!
NOT Migration Assistant or Setup Assistant, but using the "restore from backup" option in Recovery.
I was not trying to migrate data from an old computer to a new computer, however. My hard drive directories got corrupted, I couldn't fix it with the Disk Untility, so I had to reformat my hardrive and reinstall Mountain Lion from the recovery mode, with the idea of restoring my computer to the state it was before by using my my last Time Machine backup.
After reformatting and reinstalling ML, I was given the option of using the Setup Assistant, which is probably what you want to use when you first fire up a brand-spanking-new computer and want to transfer data and user accounts from an old computer. And I agree that using a Time Machine backup on an external hard drive is the best way to do this.
But I had the same problem with Setup Assistant that the OP describes with Migration Assistant – that is, it hung up when the process displayed "less than one minute left." This happened twice. When I rebooted, no data had transferred, and I was prompted to continue with the setup process.
After reading this post, the next time I rebooted, I held down the Option key and booted from the Recovery disk and chose the "Recover from backup" option. It worked like a charm.
In a couple of hours, my computer was returned to the exact state it was before the directory corruption, with the same user accounts and the most current version of Mountain Lion. I didn't have to fiddle with administrator acounts. I didn't have to update any applications or enter licenses or keys, either. Everything was current. I DID have to let Mail resync, as well as my DropBox and Google Drive accounts. And Aperture had to rebuilt its library. Photoshop and several other apps fired right up, though. All of the internet browsers I use even remembered my settings and I didn't have to re-log in to any websites I've saved my passwords for.
This might not work if you are trying to transfer data from an older version of OS X or a PC. I think that's what Migration Assistant is for. I'm not sure. Might be worth a shot, though, if Migration or Setup Assistant keeps hanging up.
Thanks, VajnaiD!