Assuming it's SATA, yeah, just buy a new external hard drive case and crack open the old Seagate enclosure (it's OK if you break it open, since you will not be re-using it, sometimes the clips snap off and break).
I saved my sister's data the same way. It would not recognize the drive and she thought it was dead. I bought a new enclosure popped it in there and her data remained intact.
It looks like the new enclosure the guys recommend above is USB 3.0 as well which should speed things up.
Good luck!
DeL