Point source detection at low signal-to-noise ratio (SNR) is challenging for astronomical surveys, particularly in radio interferometry images where the noise is correlated. Machine learning is a promising solution, allowing the development of algorithms tailored to specific telescope arrays and science cases. We present DEEPSOURCE - a deep learning solution - that uses convolutional neural networks to achieve these goals. DEEPSOURCE enhances the SNR of the sources in the original map and then uses dynamic blob detection to detect sources. Trained and tested on two sets of 500 simulated 1◦ × 1◦ MeerKAT images with a total of 300 000 sources, DEEPSOURCE is essentially perfect in both purity and completeness down to SNR = 4 and outperforms PYBDSF in all metrics. For uniformly weighted images, it achieves a Purity × Completeness (PC) score at SNR = 3 of 0.73, compared to 0.31 for the best PYBDSF model. For natural weighting, we find a smaller improvement of ∼40 per cent in the PC score at SNR = 3. If instead we ask where either of the purity or completeness first drops to 90 per cent, we find that DEEPSOURCE reaches this value at SNR = 3.6 compared to the 4.3 of PYBDSF (natural weighting). A key advantage of DEEPSOURCE is that it can learn to optimally trade off purity and completeness for any science case under consideration. Our results show that deep learning is a promising approach to point source detection in astronomical images.