Hi,

If Norton is old and no longer receive virus definitions because the subscription has expired, it is best to remove it, and then use another antivirus protection.

If you want to remove Norton and there's no other Norton product in your computer, use the Norton Removal Tool which you can download, for free at:
http://us.norton.com/support/kb/web_view.jsp?wv_type=public_web&docurl=20080710133834EN

Not sure what edition of Norton you got, but Norton Removal tool will remove Norton 2003 and higher edition. If you have Norton 2002 and ealier, you need to try removing it using Add/Remove Programs.