The ASA firewall can not route traffic. You need to masq the inside address against the external address.
Solution 1: DNS doctoring with static NAT
Let's say your external website IP address is 18.104.22.168, wich then again is port-forwarded (or directly NAT'ed) to the internal IP address 192.168.0.10. With DNS doctoring, the following will happen:
- The client on the inside requests http://www.companyweb.com, wich originally translates to 22.214.171.124
- The ASA intercepts the DNS reply packet, and replaces the A-record with 192.168.0.10
- The client gets very happy, as it now can open up the company web site :-)
For more detailed info on how you enable this: http://www.cisco.com/en/US/products/ps6120/products_configuration_example09186a00807968d1.shtml
Solution 2: Internal DNS server
This one is useful if you only have one external IP, and you port-forward this IP to many internal services on different servers (Let's say port 80 and 443 goes to 192.168.0.10, port 25 goes to 192.168.0.11 etc).
It requires no configuration change on the ASA, but it will require you to duplicate your external domain on a internal DNS server (Active Directory has this built in). You just create the exact same records as you have now, only with internal IP's on the services you have internally.
"Solution" 3: DMZ interface with public IP's
I'm not going to get into much details on this one, as it requires you to get a subnet of IP adresses from your ISP routed to your ASA. It's very hard these days with the IPv4 starvation.