I found this great resource that stepped me through the process:
How to Move or Copy an SSL Certificate from one Server to Another
Turns out it was super easy. The important things to remember are:
- The name must be exactly the same (such as mysite.myplace.com)
- The original certificate issued by the SSL provider must have the private key marked as exportable
It also helped that the old and new Windows servers used the same version of IIS.