Objective. Spike sorting, i.e. the detection and separation of measured action potentials from different extracellularly recorded neurons, remains one of the bottlenecks in deciphering the brain. In recent years, the application of neural networks (NNs) for spike sorting has garnered significant attention. Most methods focus on specific sub-problems within the conventional spike sorting pipeline, such as spike detection or feature extraction, and attempt to solve them with complex network architectures. This paper presents DualSort, a simple NN that gets combined with downstream post-processing for real-time spike sorting. It shows high efficiency, low complexity, and requires a comparatively small amount of human interaction. Approach. Synthetic and experimentally obtained extracellular single-channel recordings were utilized to train and evaluate the proposed NN. For training, spike waveforms were labeled with respect to their associated neuron and position in the signal, allowing the detection and categorization of spikes in unison. DualSort classifies a single spike multiple times in succession, as it runs over the signal in a step-by-step manner and uses a post-processing algorithm that transmits the network output into spike trains. Main results. With the used datasets, DualSort was able to detect and distinguish different spike waveforms and separate them from background activity. The post-processing algorithm significantly strengthened the overall performance of the model, making the system more robust as a whole. Although DualSort is an end-to-end solution that efficiently transforms filtered signals into spike trains, it competes with contemporary state-of-the-art technologies that exclusively target single sub-problems in the conventional spike sorting pipeline. Significance. This work demonstrates that even under high noise levels, complex NNs are not necessary by any means to achieve high performance in spike detection and sorting. The utilization of data augmentation on a limited quantity of spikes could substantially decrease hand-labeling compared to other studies. Furthermore, the proposed framework can be utilized without human interaction when combined with an unsupervised technique that provides pseudo labels for DualSort. Due to the low complexity of our network, it works efficiently and enables real-time processing on basic hardware. The proposed approach is not limited to spike sorting, as it may also be used to process different signals, such as electroencephalogram (EEG), which needs to be investigated in future research.