Measurement of the Galactic neutral atomic hydrogen (HI) column density, NHI, and brightness temperatures, Tb, is of high scientific value for a broad range of astrophysical disciplines. In the past two decades, one of the most-used legacy HI datasets has been the Leiden/Argentine/Bonn Survey (LAB). We release the HI 4-pi survey (HI4PI), an all-sky database of Galactic HI, which supersedes the LAB survey. The HI4PI survey is based on data from the recently completed first coverage of the Effelsberg-Bonn HI Survey (EBHIS) and from the third revision of the Galactic All-Sky Survey (GASS). EBHIS and GASS share similar angular resolution and match well in sensitivity. Combining both, they are ideally suited to be a successor to LAB. The new HI4PI survey outperforms the LAB in angular resolution (16.1', FWHM) and sensitivity (43mK RMS). Moreover, it has full spatial sampling and thus overcomes a major issue of LAB, which severely undersamples the sky. We publish all-sky column density maps of the neutral atomic hydrogen in the Milky Way, along with full spectroscopic data, in several map projections including HEALPix.