Current control of grid-connected converters may result in harmonic instability when grid impedance changes. To prevent this issue, current controller parameters can be tuned adaptively according to different short-circuit ratios (SCRs). It is thus important to estimate the grid impedance in real-time. Unlike traditional FFT-based impedance measurement methods, a more efficient estimation approach based on neural networks is proposed in this paper. This method does not require a fixed and relatively long sampling window, making it possible for real-time impedance measurement. Further, a step-by-step design method of the feedforward neural network (FNN) used for grid impedance estimation is developed. Time-domain simulation results validate the effectiveness of the approach. Based on the designed FNN, adaptive current control is implemented and verified through simulation.