This letter presents a novel system to perform range detections using an artificial neural network on the time-domain baseband signal of frequency-modulated continuous wave radar sensors. The network is trained and evaluated with synthetic signals, which are generated with a point target simulator. To evaluate the performance of the proposed approach, it is compared with an order statistics constant false alarm rate (CFAR) detector at different signal-to-noise ratios. The detection system is shown to work-in some cases even outperforming the baseline-in synthetic single-target, as well as in multiple-target scenarios. Therefore, it is capable of replacing the usual fast Fourier transform and CFAR detection procedures in radar signal processing. Furthermore, it is demonstrated that the detection system also works with real radar measurement data.