A comprehensive catalog of historical earthquakes, with accurate epicenters and homogenized magnitudes is a crucial resource for seismic hazard mapping. Here we update and combine catalogs from several sources to compile a catalog of earthquakes in and near Iceland, in the years 1900–2019. In particular the epicenters are based on local information, whereas the magnitudes are based on teleseismic observations, primarily from international on-line catalogs. The most reliable epicenter information comes from the catalog of the Icelandic Meteorological Office, but this is complemented with information from several technical reports, scientific publications, newspaper articles, and modified by some expert judgement. The catalog contains 1272 $M_W \geq 4$ events and the estimated completeness magnitude is $M_W$ 5.5 in the first years, going down to $M_W$ 4.5 for recent years. The largest magnitude is $M_W$ 7.01. Such melting of local and teleseismic data has not been done before for Icelandic earthquakes, and the result is an earthquake map with no obviously mislocated events. The catalog also lists additional 5654 earthquakes on the Mid-Atlantic Ridge, north of 43\degree, with both epicenters and magnitudes determined teleseismically. When moment magnitudes are not available, proxy $M_W$ values are computed with $\chi^2$-regression, normally on $M_S$, but exceptionally on $m_b$. All the presented magnitudes have associated uncertainty estimates. The actual combined seismic moment released in the Icelandic earthquakes is found to be consistent with the moment estimated using a simple plate motion model. The catalog is named ICEL-NMAR and it is available online at http://dx.doi.org/10.17632/tg67sphksh.1.