There have been several trans-Atlantic crossings by replica Viking longboats in the past.
Norwegians built one in Norway and sailed it to the Chicago’s World Columbia Exposition (Chicago World Fair) in 1893. The Exposition was to celebrate the 400th anniversary of Columbus’s trans-Atlantic crossing.
http://www.vikingship.us/history.htm
Then in the 1920s another Norwegian, a merchant mariner, had a longboat built, and sailed it from Norway to Boston. That boat has been on display in Duluth, MN, and is presently under restoration.
http://leiferiksonvikingship.com/folgeros_voyage.htm
There was also a replica longboat built in a Minnesota potato shed about 45 years ago, and ultimately sailed eastbound across the Atlantic. That vessel is on display in Minnesota.
http://www.hcscconline.org/current-exhibits/hjemkomst-viking-ship/
This current project must not be getting much press in Norway. My Norwegian relatives, active sailors on the Oslofjord and beyond, had not heard of it over there.