To glean the benefits offered by massive multi-input multi-output (MIMO) systems, channel state information must be accurately acquired. Despite the high accuracy, the computational complexity of classical linear minimum mean squared error (MMSE) estimator becomes prohibitively high in the context of massive MIMO, while the other low-complexity methods degrade the estimation accuracy seriously. In this paper, we develop a novel rank-1 subspace channel estimator to approximate the maximum likelihood (ML) estimator, which outperforms the linear MMSE estimator, but incurs a surprisingly low computational complexity. Our method first acquires the highly accurate angle-of-arrival (AoA) information via a constructed space-embedding matrix and the rank-1 subspace method. Then, it adopts the post-reception beamforming to acquire the unbiased estimate of channel gains. Furthermore, a fast method is designed to implement our new estimator. Theoretical analysis shows that the extra gain achieved by our method over the linear MMSE estimator grows according to the rule of O($\log_{10}M$), while its computational complexity is linearly scalable to the number of antennas $M$. Numerical simulations also validate the theoretical results. Our new method substantially extends the accuracy-complexity region and constitutes a promising channel estimation solution to the emerging massive MIMO communications.