The current C network - is it a single class C with no subnetting, that you intend to migrate to multi segment network (i.e. core + servers + clients + printer subnets)? How many switches does all of the current network reside on, and are you doing any network upgrades at the same time?
One mode if migration if you have the possibility and a few extra switches, is to allocate a new core infrastructure, and on it declare the old C network (s) as subnets hanging off one port of the new/reallocated core. Then you can move client switches to the new addressing one at a time. Servers are moved to the new core (or new server farm switch) one at a time, or during several evening/weekend migrations.
Eventually the old core and subnet is retired / reallocated or set up for redundancy with the new one.
This allows you to migrate slowly; or you can still do it in a weekend or two; but does allow you to "experience" any issues prior to a big bang.
Internet connection and WAN connections can proceed with the above; or even stay with the old C address if you don't have to give it up.
Caveats:
1. Printers - they might be more work than you think. In client subnet, or a dedicated printer one?
2. Windows servers, Active directory, and DNS. Sometimes there's a replication time involved.
3. Internet and PAT - don't forget it ! Firewall rules may need extensive overhaul.
4. Licensing, if any applications are tied to IP addresses.
5. Make sure to have access to all the little oddball devices you have out there - cameras, power monitoring, modems, security systems, network management, PBX, the president's printer...
