I wonder how generally work the IP addresses on dedicated servers providers? The IP addresses are generally rented so you have no option to keep the IP's if you move to another provider? I see that many of those companies allow you to change some information in the IP WHOIS databases but that doesn't mean that you own the IP?
If you're sticking with the same provider, in some cases they can reassign them to the new service, but there's no guarantees, and they may charge you some serious coin to make it happen.
If you're going with a new provider, you can't keep the IPs (unless they're assigned to you, which I'm going to go out on a limb and say they're not).
But it shouldn't be a problem. If you've done your network design right, renumbering isn't painful, it just takes a little bit of attention to detail. Of course, if your network design is a dog's breakfast, you're in for a world of pain, but you can use that as a learning experience for next time. Specific points to note include:
Renumbering would be significantly easier, too, if some [expletive deleted][expletive deleted][expletive deleted][expletive deleted][expletive deleted][expletive deleted] caching DNS resolvers didn't override your DNS TTL values. Given that there is absolutely nothing you can do about it, you just need to have a plan to work around it.
First, do the right thing with DNS TTLs so that the well-behaved majority don't take a hit. Drop your TTLs to something nice and small (like 5 minutes) at least twice the current TTL before the cutover (why twice the current TTL? That way if you do it wrong, you can notice in time and fix it).
Make sure the contract on your old server overlaps your cutover date by at least a month. When you cutover to the new site, insert some NAT rules on your old server to redirect traffic to the new server. Only do this for selected ports (if you do the whole machine, you'll lock off your ability to get into it later, and you don't want to pay for all of the gibberish traffic -- portscans and the like -- that are the whitenoise of your traffic graphs). Keep an eye on your logs for connections from your old IP address on the new server -- when they've dropped to nothing, then the world has forgotten your old IP address, and you can drop the NAT rules on the old server and decommission it.
IP addresses are NOT rented, they are assigned. In blocks of at least 4096 which are not splittable.
You want your own?
Otherwise, sorry - the IP addreess is part of an AS assigned block, and at the end of the day this is not splittable.