I have only done this once from scratch at home through a broadband modem connection, but it worked well and seemed to be stable.
1) Register your domain. For my home use, I registered with Register.com, $35 per year, and they host your DNS entries for you, and allow you to manipulate your DNS entries though a web interface. The flip side is to register your name with an entity and have them point at your DNS server on site, more maintenance for you because you need to have a DNS server locally. There are a bunch of registration companies to choose from, and I'm sure you can find the same service for less if you look around.
2) In your DNS, you will need to set up an MX record and put in the IP address of your mail server. Do you have a static IP for your mail server? If so, use that. If not, used the DNS entry that the broadband provider supplies for your gateway connection out to the 'net.
3) If you have a broadband connection and some sort of router, or firewall (highly recommended), you'll have to allow traffic through that device on port 25 to the mail server. And if you are doing NAT through your gateway, you will have to redirect port 25 from your external IP to whatever your internal private IP is.
4) Install the Internet Mail Connector in Exchange, or if it's installed, configure it to listen for mail. After the config, it will run as the "Microsoft Exchange Internet Mail Service" under Services.
5) Test it. From an external IP address, do a Telnet yourserver.yourdomain.??? 25 and you should get a reply similar to "ESMTP Server (Microsoft Exchange Internet Mail Service 5.5.2653.13) ready". It's listening for mail at that point.
Like I said, only did this once, so I may have left something out. Also, any constructive criticisms would be appreciated.
Doug