yeah, there shouldnt be any problems cuz ur not runnin the OS and ur not ghosting ur drive, ur simply installing it as a slave. just MAKE SURE u set ur jumpers (the current drive in ur new computer needs to be on MASTER as well, NOT THE C/S setting). then ur drive that ur installing should be the SLAVE drive. if it doesnt say on the harddrive itself (it usually does), u can do a simple search online depending on which manufacturer drive u have and find out the orientation of the jumpers.
one other thing that wasnt really mentioned, which SHOULDNT cause a big problem or a problem at all, is the powersupply in the computer. i work in retail sales on manufacturer sales for HP, Gateway, EMachines, and Compaq, and one of the things that people dont realize is that the powersupplies inside those computers are as MINIMALISTIC as possible (cheap manufacturers -_- that provide those computers with powersupplys with like 250W-300Ws of power...). so if u find the new drive is giving u stability problems, then i would recommend upgrading ur powersupply to something with a little more juice.