From the Wikipedia page [1]

The National Archives of Iceland is the national archive of Iceland, located in Reykjavík.

Founded in 1882 by the Danish appointed governor (landshøvding) of Iceland, Hilmar Finsen, the National Archives of Iceland (NAI) drew its material mainly from governmental archives. At first, it was located in the attic of the Cathedral of Reykjavík. The National Library of Iceland and the National Museum of Iceland had been moved from the loft of the Cathedral to the new Parliament Building (Alþingishús), and the governor took this opportunity to use the Cathedral loft for the new archives, so bringing together the archive materials that had accumulated in various government departments and providing them with better facilities for preservation. The archives have moved to new premises three times since: first to the Alþingishús in 1900, then to the new Archives and Library building on Hverfisgata in 1908 and finally to their present location at Laugavegur 162 in 1987-98.

The website, in Icelandic, is [2]