This paper introduces a method based on a deep neural network (DNN) that is perfectly capable of processing radar data from extremely thinned radar apertures. The proposed DNN processing can provide both aliasing-free radar imaging and super-resolution. The results are validated by measuring the detection performance on realistic simulation data and by evaluating the Point-Spread-function (PSF) and the target-separation performance on measured point-like targets. Also, a qualitative evaluation of a typical automotive scene is conducted. It is shown that this approach can outperform state-of-the-art subspace algorithms and also other existing machine learning solutions. The presented results suggest that machine learning approaches trained with sufficiently sophisticated virtual input data are a very promising alternative to compressed sensing and subspace approaches in radar signal processing. The key to this performance is that the DNN is trained using realistic simulation data that perfectly mimic a given sparse antenna radar array hardware as the input. As ground truth, ultra-high resolution data from an enhanced virtual radar are simulated. Contrary to other work, the DNN utilizes the complete radar cube and not only the antenna channel information at certain range-Doppler detections. After training, the proposed DNN is capable of sidelobe- and ambiguity-free imaging. It simultaneously delivers nearly the same resolution and image quality as would be achieved with a fully occupied array.