Non-invasive fetal electrocardiography has the potential of providing vital information for evaluating the health status of the fetus. However, the low signal-to-noise ratio of the fetal electrocardiogram (ECG) impedes the applicability of the method in clinical practice. Residual noise in the fetal ECG, after the maternal ECG is suppressed, is often non-stationary, complex and has spectral overlap with the fetal ECG. In this paper, we propose the use of artificial intelligence for removing the residual noise from single-channel fetal ECG. We present a deep fully convolutional encoder-decoder framework, learning end-to-end mappings from noise-contaminated fetal ECG signals to clean ones. The encoder acts as a feature extractor that eliminates the noise while preserving the primary ECG components, whereas the decoder recovers the signal details. Symmetric skip-layer connections pass information from convolutional to transposed convolutional layers aiding in restoring the clean fetal ECG. The method was tested in a broad simulated fetal ECG dataset with varying amount of noise. The results demonstrate that after the denoising there was an average increase in the correlation coefficient between the corrupted signals and the original ones from 0.6 to 0.8. Moreover, the suggested framework successfully handled different levels of noises in a single model. The network was further tested on real signals showing substantial noise removal performance, thus providing a promising approach for fetal ECG signal denoising. The presented method is able to significantly improve the quality of the extracted fetal ECG signals, having the advantage of preserving beat-to-beat morphological variations.