If you're using VMWare, there is a tool provided to virtualize existing systems. Basically it makes a copy of the old system (while it is down) and recreates it as a VM with the same name, IP, etc. I haven't double-checked to see if this process is 'officially' supported by SAP, and I don't know the technical details about how it's done or how long it will take (VMWare is managed by the Network Operations group in my team, so as the SAP sysadmin I'm basically a customer of theirs). However, we have virtualized a few smaller systems that needed to get off failing hardware in a hurry and it worked very well, so I would expect it would for an SAP dialog instance as well. As you're not copying a database, just the app server, it probably would fit within your maintenance window.
If this route works for you, then there is no need to actually install the SAP system on the VM instance... you just copy the old server as-is, then make any adjustments required due to no longer needing old drivers, etc, plus potentially having different memory parameters.