Archiving projects so that their tasks no longer show up in Timesheets

I am looking for a way to archive projects such that employees can no longer charge time against their tasks. As far as I can see the only way to do this is to delete projects to the Recycling Bin, which is too risky to lose and not the right place for archives. 

One thing that would work is creating a Space called Archive where completed projects are moved to.  Then archive the space (un-archive when a project needs to be moved there)  This would work for me so long as Tasks that are in the Archived space don't show up in  "Add Task" in Timesheets, but they currently do. 

any help would be greatly appreciated!