name on the security certificate is invalid



I set up an SSL Certificate for my website. All works fine when the
site is accessed via www.mysite.com.

The problem is that there are many internal users and I don't want
them to have to go out to the internet to access the site. They
access the site with http://servername. This causes the error "name
on the security certificate is invalid".

What's the best solution for supporting both internal and external
customers?

More info:
1. For the internal users, the connection doesn't necessarily need to
be encrypted
2. On the website, I have "require ssl" checked (in IIS). Then, I
use a custom error page to redirect people to https if the come in to
the sive with http. I'd like to keep this part. It works well.
3. Is there a way to allow internal users to type www.mysite.com and
have the router (cheap home router) keep the traffic inside the
building without going to the ISP and back?
4. Is there a way to install a second cert with the server name?
5. I guess I could maintain two copies of the site - one for internal
and one for external customer... but I don't like this solution. Does
anyone else do this?
6. Is there a way to make a single cert work for www.mysite.com and
"http://myservername";? What about when people hit the site using just
an IP address (internal or external)?

That's it for now - any help would be greatly appreciated.

Thanks.

.