Should be straight forward enough. I suggest you buy an enclosure (Caddy) which will be useful to use the old HDD as an external hard disk for back ups. click here
The other option is to borrow another PSU for your existing machine to get it working temporarily and use Vista`s Windows Easy Transfer to move all your data and settings to the new computer. You could save the data file to the existing HDD, then put it in the caddy and transfer from there to the new Vista Machine.