You will need to add the *.domain.domainextension to the trusted internet zone in IE (Push this through Group Policy) as recommended by Colin. This will stop the prompts for browsing but will NOT fix authentication problems with the Office SharePoint Sites library (MySite, Member Sites) for Vista and Windows 7 machines since the WebClient service does not use the same list of trusted sites to determine Intranet or external links.
I've written a guide based on experiences with this problem on my blog here: http://thechriskent.com/2012/03/02/fixing-sharepoint-office-integration-problems/
Basically you will need to add a Multi-String Value under the HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\WebClient\Parameters key called AuthForwardServerList and set the value to http://*.domain.domainextension