Church History Library
The 250,000 square foot Church History Library, now under construction, is registered with the US Green Building Council. (source) To the best of my knowledge, this is the first building owned by the LDS Church to be registered for a Green Building certification.
This doesn’t guarantee any specific rating, but simply means that the paperwork and money have been submitted. Once construction is completed, the point total will determine the overall rating achieved, with a basic Certification being the lowest and Platinum being the highest. Originally to be finished in late 2007, the new library will be completed in summer 2009.