Massive MIMO is widely touted as an enabling technology for 5th generation (5G) mobile communications and beyond. On paper, the large excess of base station (BS) antennas promises unprecedented spectral efficiency gains. Unfortunately, during the initial phase of industrial testing, a practical challenge arose which threatens to undermine the actual deployment of massive MIMO: user mobility-induced channel Doppler. In fact, testing teams reported that in moderate-mobility scenarios, e.g., 30 km/h of user equipment (UE) speed, the performance drops up to 50% compared to the low-mobility scenario, a problem rooted in the acute sensitivity of massive MIMO to this channel Doppler, and not foreseen by many theoretical papers on the subject. In order to deal with this "curse of mobility", we propose a novel form of channel prediction method, named Prony-based angular-delay domain (PAD) prediction, which is built on exploiting the specific angle-delay-Doppler structure of the multipath. In particular, our method relies on the high angular-delay resolution which arises in the context of 5G. Our theoretical analysis shows that when the number of base station antennas and the bandwidth are large, the prediction error of our PAD algorithm converges to zero for any UE velocity level, provided that only two accurate enough previous channel samples are available. Moreover, when the channel samples are inaccurate, we propose to combine the PAD algorithm with a denoising method for channel estimation phase based on the subspace structure and the long-term statistics of the channel observations. Simulation results show that under a realistic channel model of 3GPP in rich scattering environment, our proposed method is able to overcome this challenge and even approaches the performance of stationary scenarios where the channels do not vary at all.