It's possible, but annoying. basically there's no way to hook two hard drives up together directly, but you can transfer files between two differnt configurations as long as they're both hooked up to a PC. basically all you need is an external hard drive enclosure. just keep the IDE in the case hooked to the motherboard, then get a SATA compatable enclosure with either firewire or USB (depending on what you use). Then just hook up the SATA drive through the enclosure to the computer, and swap the files. then disconnect both drives, and swap them inside the case.
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http://www.newegg.com/Product/ProductList.asp?N=2000090092+1053807123+1053907126+1054107131+1054207132&Submit=ENE&SubCategory=92)
there's some examples of SATA compatable enclosures with USB 2.0 connections. I'd refine your own search to what you need, cause i'm just assuming.
now that's just for transferring files. considering you wanna move the OS you'll need some tool that copies the entire physical hard drive to another. i can't think of the tool at the moment, but i bet you can find one easily enough. Just copy the entire hard drive to the other, then swap inside the case, then reboot. i dont think there's much else you gotta do.