Speech brain-computer interfaces (BCIs) have the potential to restore rapid communication to people with paralysis by decoding neural activity evoked by attempted speaking movements into text or sound. Early demonstrations, while promising, have not yet achieved accuracies high enough for communication of unconstrainted sentences from a large vocabulary. Here, we demonstrate the first speech-to-text BCI that records spiking activity from intracortical microelectrode arrays. Enabled by these high-resolution recordings, our study participant, who can no longer speak intelligibly due amyotrophic lateral sclerosis (ALS), achieved a 9.1% word error rate on a 50 word vocabulary (2.7 times fewer errors than the prior state of the art speech BCI2) and a 23.8% word error rate on a 125,000 word vocabulary (the first successful demonstration of large-vocabulary decoding). Our BCI decoded speech at 62 words per minute, which is 3.4 times faster than the prior record for any kind of BCI and begins to approach the speed of natural conversation (160 words per minute). Finally, we highlight two aspects of the neural code for speech that are encouraging for speech BCIs: spatially intermixed tuning to speech articulators that makes accurate decoding possible from only a small region of cortex, and a detailed articulatory representation of phonemes that persists years after paralysis. These results show a feasible path forward for using intracortical speech BCIs to restore rapid communication to people with paralysis who can no longer speak.