If I'm understanding this correctly, you'll need to set up the DMZ on the router, because of the PIX 506E only having 2 Ethernet interfaces. The PIX 515E is the first model to have additional interfaces, allowing for dedicated DMZ port/access.
If you have two available ethernet ports on your 1720, you could use one of them for your DMZ, connecting a switch/hub to this port, and placing your publicly accessible servers on this switch/hub.
This link gives a diagram of what your scenario might look like in the end...
http://www.cisco.com/en/US/partner/products/sw/secursw/ps1018/products_configuration_example09186a0080094111.shtml
The use of the IOS Firewall Feature Set on the router, when placing a DMZ on it, is a good idea. Look into that further as well.
As far as public/private ip address concerns, you can use NAT on a DMZ interface if you'd like, adding a small form of security to the DMZ network.
Hopefully this helps a bit...