We present the first reconstruction of dark matter maps from weak lensing observational data using deep learning. We train a convolution neural network with a U-Net-based architecture on over 3.6 × 10^5 simulated data realizations with non-Gaussian shape noise and with cosmological parameters varying over a broad prior distribution. We interpret our newly created dark energy survey science verification (DES SV) map as an approximation of the posterior mean P(κ|γ) of the convergence given observed shear. Our DeepMass^1 method is substantially more accurate than existing mass-mapping methods. With a validation set of 8000 simulated DES SV data realizations, compared to Wiener filtering with a fixed power spectrum, the DeepMass method improved the mean square error (MSE) by 11 per cent. With N-body simulated MICE mock data, we show that Wiener filtering, with the optimal known power spectrum, still gives a worse MSE than our generalized method with no input cosmological parameters; we show that the improvement is driven by the non-linear structures in the convergence. With higher galaxy density in future weak lensing data unveiling more non-linear scales, it is likely that deep learning will be a leading approach for mass mapping with Euclid and LSST.