I'm thinking you can't move the entire subnet that the server is in (which would be the easy option). So assuming that is not the case ... I would personally setup a DNS entry pointing to the server, and update all the remote sites to use the DNS entry. Then it would be easy to change. Another option would be to setup an HAProxy using the current server's IP address and have HAProxy forward the request to the Azure VM. Then start slowly updating the sites to use the new Azure VM IP address. This will allow the clients to use both the old and new IP address at the same time while you transition them across.
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