Setup New Domain Name, Sub-directory install

I have a fully working multisite wordpress install. So far so good.
Over the last few weeks I added another site that comes up in a browser when you type:

http://www.myserverbox.com/sitename

Now I have a domain name

http://www.actualsitename.com

The wordpress install is in the root of a linux server that has a fixed IP address and you can access this at

http://www.myserverbox.com

That gets you to the main root site/admin/etc.

I have the ability to add IP addresses and domain name records to the box. In fact I have another IP and website (not wordpress related) working just fine.

All I would like to do is have the outside world be able to type

http://www.actualsitename.com

and reach

http://www.myserverbox.com/sitename

it would GREAT if it would also translate links to

http://www.actualsitename.com/LINK

as well.

I have complete access to the Domain records and the wordpress install and can change anything I need to.

Please help!!

Thank you,

Rich