I have a family member who has the same last name as Justice Antonin
Scalia.  This family member would like to know if she is related to
the Justice.  Is there a public e-mail address where he can be
reached?

I would suggest that you call Justic Scalias office.  They can be
reached at (202) 479-3211.  As you would expect there is no 'offical'
to contact the Justice himself directly.

If you are more interested in snailmail:

Justice Antonin Scalia
c/o US Supreme Court 1
1st Street N.E. 
Washington, DC. 20543 

Another option is using a outside service to make contact.
http://contactanycelebrity.com specializes in these things.

Hope this is of some help.

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mediatek01