We construct the largest curved-sky galaxy weak lensing mass map to date from the DES first-year (DES Y1) data. The map, about 10 times larger than the previous work, is constructed over a contiguous approximate to 1500 deg(2), covering a comoving volume of approximate to 10 Gpc(3). The effects of masking, sampling, and noise are tested using simulations. We generate weak lensing maps from two DES Y1 shear catalogues, METACALIBRATION and IM3SHAPE, with sources at red-shift 0.2 < z < 1.3, and in each of four bins in this range. In the highest signal-to-noise map, the ratio between the mean signal to noise in the E-mode map and the B-mode map is similar to 1.5 (similar to 2) when smoothed with a Gaussian filter of sigma(G) = 30 (80) arcmin. The second and third moments of the convergence kappa in the maps are in agreement with simulations. We also find no significant correlation of kappa with maps of potential systematic contaminants. Finally, we demonstrate two applications of the mass maps: (1) cross-correlation with different foreground tracers of mass and (2) exploration of the largest peaks and voids in the maps.