I would like to run a Windows Service under the identity of a User Account that exists on a different Active Directory Domain than the one the host machine is currently joined to.
What kind of configuration is required? Do the two domains need to have a trust relationship?
You would need to setup a trust relationship between the two domains. Then you would need to grant the user from the other domain the correct rights to run the service as well as the 'Run as a Service' Right on the box you will be running the service on.
I would note even if the user is a domain admin on it's "home domain" it will only be a regular user on your domain so design your rights appropriately.