What was the first U.S. government building constructed?

Your question is somewhat ambiguous. Are you asking about the first building used by the U.S. government? That's probably Carpenter's Hall in Philadelphia where the First Continental Congress met. It's still being used, but not as a government building.

Apparently the oldest U.S. government building still used by the government is the Capitol in Washington, D.C.

Check this site for more details.

http://en.wikipedia.org/wiki/United_States_Capitol#Previous_Capitols

Thanks a lot!

Holly = )