My quick google research came up with this, so I don't know how they could have had a website in the 80's.
August 6, 1991
The first web page went live on August 6, 1991. It was dedicated to information on the World Wide Web project and was made by Tim Berners-Lee. It ran on a NeXT computer at the European Organization for Nuclear Research, CERN. The first web page address was http://info.cern.ch/hypertext/WWW/TheProject.html