WordPress Multisite: add a new site with domain name

First, follow all instructions on for setting up a network of WP sites. These instructions can be found here: WordPress Codex: create a nework.

Then, install the WP MU Domain Mapping plugin.

Step 1: add a new site

Now you’re ready to add a site and map it to your domain. Go to My Sites > Network Admin > Dashboard > Sites and Click on Add New. Here you’ll have to write something  in the “Site Address” field – this will be “reverted” in Step 3.

Step 2: domain mapping

Go to My Sites > Network Admin > Dashboard > Settings > Domains. If your new site is your first “sub-site” in the network, ID will be 2, and so on.

Step 3: real domain name instead of subdomain

Now go back to My Sites > Network Admin > Dashboard > Sites and edit your site. You can now change the “Domain” field to match the real domain name of your site.

Links:

Here are the two most useful article referring to adding new domains to WP Network and mapping them to domain names:
How to map domains to sites within WordPress multisite
WordPress Multisite Beyond Basics: Essentials and Domain Mapping