The purpose of this research was to implement a deep learning network to overcome two of the major bottlenecks in improved image reconstruction for clinical positron emission tomography (PET). These are the lack of an automated means for the optimization of advanced image reconstruction algorithms, and the computational expense associated with these state-of-the art methods. We thus present a novel end-to-end PET image reconstruction technique, called DeepPET, based on a deep convolutional encoder-decoder network, which takes PET sinogram data as input and directly and quickly outputs high quality, quantitative PET images. Using simulated data derived from a whole-body digital phantom, we randomly sampled the configurable parameters to generate realistic images, which were each augmented to a total of more than 291,000 reference images. Realistic PET acquisitions of these images were simulated, resulting in noisy sinogram data, used for training, validation, and testing the DeepPET network. We demonstrated that DeepPET generates higher quality images compared to conventional techniques, in terms of relative root mean squared error (11%/53% lower than ordered subset expectation maximization (OSEM)/filtered back-projection (FBP), structural similarity index (1%/11% higher than OSEM/FBP), and peak signal-to-noise ratio (1.1/3.8 dB higher than OSEM/FBP). In addition, we show that DeepPET reconstructs images 108 and 3 times faster than OSEM and FBP, respectively. Finally, DeepPET was successfully applied to real clinical data. This study shows that an end-to-end encoder-decoder network can produce high quality PET images at a fraction of the time compared to conventional methods.