Is it possible to have an external hard drive act as the internal hard drive?
Yes and no. It depends on whether the computer can boot from a USB hard drive.
I wouldn't recommend it, though, as the USB bus will only slow you down.
I guess you may want to ask yourself if it's worth buying "duct-tape fixes" for your computer to get it to recognize a SATA drive (i.e.: an IDE to SATA connector, which you found didn't work; a PCI SATA controller card, which may or may not work, depending on whether the motherboard's BIOS can boot from a PCI controller card).
If you really need 640 GB, get an external hard drive and DO NOT use it as a boot drive. Use it as a secondary hard drive for storage.
Otherwise, get the largest IDE drive you can get.
...or save up the cash and get a new computer...