To expand on the above, if the old case doesn't have a sticker, it's all a bit academic because you need the Product Key. It will be an OEM system, branded to the machine it was installed on and as such you can't transfer it, even with the Dell range. Does the replacement not have a genuine Vista system?
It's probably best to splash out for a new and genuine Microsoft disk and key for Windows 7. Vista will be next in line for destruction now that XP has been given its burn notice.