Do you mean to say that you are hosting a server with your ISP? If so, there isn't much else you can do if you're switching ISPs, sorry.
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Before making any changes of dns, configure everything for the other ip address. Once you're set on that new ip, make your dns change, whilst keeping your website alive on the old ip address.
All of your users should make the switch to the new ip address within a few days. If you find that they are all using the old address, provide them with directions for editing their c:\windows\system32\drivers\etc\HOSTS file, or if they are using linux / *bsd their /etc/hosts file.
You can't transfer ip addresses unless the ip addresses you host on are transportable.
You will be assigned new ip addresses by your new ISP. It'd be a good idea to just keep both connections alive until all your users have made the switch to the new ISP. Then, once all your visitors have reverted to the new ISP, you can drop the old connection.
Is this just a switch of connection, or are you switching datacenters as well? Do you have a dedicated server, or is it colocated?
I dont resell my hosting service's yet... i just host my site and a few of my freinds company sites, its about 15 websites. We are going to change ISP's becuase Echolons support sucks and they are charging alot for a partial T1, And we need to go up to a full T1.