Well since you already have DHCP, it shouldn't be a problem.
1. Configure your network for routing with new address range. Routing for
all addresses should remain in place.
2. Alter configuration of your DHCP servers so they start issuing addresses
from the new range. Configure DNS ans default gateway options and others
that you use accordingly.
3. Analyze your network evvironment to determine if you are ready to move
the servers. For example, if you are using UNC paths with ip addresses for
drive mapping, you aren't ready yet (have to remove them or change to use
server names). Moving your servers could be a challenging or a very easy
task, depending on your infrastructure complexity. Note: don't forget
changing appropriate DHCP options as server's ip addresses change!
4. Deactivate and remove old DHCP scope. When your clients renew their
address leases, you will be on the new address range completely.
5. Remove routing for the old address range.
