There has been a church in this location since at least 1200 A.D., but possibly since 1000 A.D. The current church is built of stone blocks, one of the oldest stone buildings in the country. Building began in 1773 and the church was consecrated in 1796. The tower was completed in 1823. The church interior went through extensive restoration and alterations in 1945-47.
The church stands close to the residency of the President of Iceland, the façade of which can be just glimpsed at the right edge of the photo.
Incidentally, there will be a presidential election in Iceland on June 30.
Several people are in the running, including the current incumbent, Ólafur Ragnar Grímsson.