Assuming the new IP address is on the same subnet as the first, add a second virtual interface (sometimes called an "alias") to the primary network interface. This is configured, like all network interface settings, in
/etc/network/interfaces. The Debian Reference manual has a section on the topic:
A simple example, assuming your primary network interface is
eth0 and has an ip of
192.168.1.1 and the new ip is
iface eth0 inet static
iface eth0:0 inet static
Once the appropriate settings have been added to
ifup eth0:0 to activate the new interface.
If, however, the new ip is on a different subnet, you need to either provision the ip on a physically distinct network interface or create a VLAN interface, depending on how your ISP is prepared to hand it off to you. That's a whole new topic.