The assy is only one file. You have to copy the associated part files also.
If you have more than a few Parts in the Assy. You may want to use Pack and Go.
This will grab all components and drawings if you choose and also break the link to the current location.
The free option is Pack and Go as Andreas Rhomberg already mentioned.
If you have a fairly large Workgroup PDM vault, I recommend getting a quote from your VAR to use the Migration tool provided to them for this job. Not only is it a time saver, but the quality of your data is maintained with good planning and migrating.