When I go to Settings > Web sites > Mappings (not sure if the names are exactly correct as I'm using localized version – sorry about that) I can map my own 3rd level domain like www.mydomain.com to my Google Site such as http://sites.google.com/a/mydomain.com/my-site.
However, 3rd level domain must always be specified so mapping like mydomain.com -> http://sites.google.com/a/mydomain.com/my-site is not allowed.
The workaround would be to set up some custom hosting and create a simple redirect for non-www version to the www one but isn't there a way to do that directly in Google Apps somehow? I don't want to set up some separate hosting just to support the shorter URL name.
Best Answer
It is possible simply by configuring your DNS settings for your domain. Google has support for it and it guides you through rather clearly to redirect your 2nd level domain to the 3rd level domain.
Note that you need access to change your domain's DNS settings and Google Apps settings for this to actually work.