Neural network (NN) emulators of the global 21 cm signal need an emulation error much less than the observational noise in order to be used to perform unbiased Bayesian parameter inference. To this end, we introduce 21cmLSTM—a long short-term memory (LSTM) NN emulator of the global 21 cm signal that leverages the intrinsic correlation between frequency channels to achieve exceptional accuracy compared to previous emulators, which are all feedforward, fully connected NNs. LSTM NNs are a type of recurrent NN designed to capture long-term dependencies in sequential data. When trained and tested on the same simulated set of global 21 cm signals as the best previous emulators, 21cmLSTM has an average relative rms error of 0.22%—equivalently 0.39 mK—and comparably fast evaluation time. We perform seven-dimensional Bayesian parameter estimation analyses using 21cmLSTM to fit global 21 cm signal mock data with different adopted observational noise levels, σ21. The posterior 1σ rms error is ≈three times less than σ21 for each fit and consistently decreases for tighter noise levels, showing that 21cmLSTM can sufficiently exploit even very optimistic measurements of the global 21 cm signal. We have made the emulator, code, and data sets publicly available so that 21cmLSTM can be independently tested and used to retrain and constrain other 21 cm models.