You need to set up your own name<->ip translation to the new ip, overriding the current one.
Let say your URL site is www.whatever.com
which binds to 188.8.131.52 on internet.
Let say your new server would be 184.108.40.206.
In a linux machine put a line in your /etc/hosts file to point to this new ip:
(you may to restart the nscd daemon. Also, in the file /etc/nsswitch.conf, the line "hosts: files dns" must have "files" first).
After changing /etc/hosts, ping www.whatever.com
would return results from 220.127.116.11.
In a windows machine the equivalent file is \Windows\System32\drivers\etc\hosts