We present a significantly-improved data-driven global weather forecasting framework using a deep convolutional neural network (CNN) to forecast several basic atmospheric variables on a global grid. New developments in this framework include an offline volume-conservative mapping to a cubed-sphere grid, improvements to the CNN architecture, and the minimization of the loss function over multiple steps in a prediction sequence. The cubed-sphere remapping minimizes the distortion on the cube faces on which convolution operations are performed and provides natural boundary conditions for padding in the CNN. Our improved model produces weather forecasts that are indefinitely stable and produce realistic weather patterns at lead times of several weeks and longer. For short- to medium-range forecasting, our model significantly outperforms persistence, climatology, and a coarse-resolution dynamical numerical weather prediction (NWP) model. Unsurprisingly, our forecasts are worse than those from a high-resolution state-of-the-art operational NWP system. Our data-driven model is able to learn to forecast complex surface temperature patterns from few input atmospheric state variables. On annual time scales, our model produces a realistic seasonal cycle driven solely by the prescribed variation in top-of-atmosphere solar forcing. Although it is currently less accurate than operational weather forecasting models, our data-driven CNN executes much faster than those models, suggesting that machine learning could prove to be a valuable tool for large-ensemble forecasting.