In this paper, we propose a novel complex convolutional neural network (CNN) CSI enhancer for integrated sensing and communications (ISAC), which exploits the correlation between the sensing parameters (such as angle-of-arrival and range) and the channel state information (CSI) to significantly improve the CSI estimation accuracy and further enhance the sensing accuracy. Within the CNN CSI enhancer, we use the complex-valued computation layers to form the CNN, which maintains the phase information of CSI. We also transform the CSI into the sparse angle-delay domain, leading to heatmap images with prominent peaks that can be efficiently processed by CNN. Based on the enhanced CSI outputs, we further propose a novel biased fast Fourier transform (FFT)-based sensing scheme for improving the range sensing accuracy, by artificially introducing phase biasing terms. Extensive simulation results show that the ISAC complex CNN CSI enhancer can converge within 30 training epochs. The normalized mean square error (NMSE) of its CSI estimates is about 17 dB lower than that of the linear minimum mean square error (LMMSE) estimator, and the bit error rate (BER) of demodulation using the enhanced CSI estimation approaches that with perfect CSI. Finally, the range estimation MSE of the proposed biased FFT-based sensing method approaches that of the subspace-based sensing method, at a much lower complexity.