Coronavirus disease 2019 (COVID-19), caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus, leads to respiratory symptoms that can be fatal. However, neurological symptoms have also been observed in some patients. The cause of these complications is currently unknown. Here, we use human-pluripotent-stem-cell-derived brain organoids to examine SARS-CoV-2 neurotropism. We find expression of viral receptor ACE2 in mature choroid plexus cells expressing abundant lipoproteins, but not in neurons or other cell types. We challenge organoids with SARS-CoV-2 spike pseudovirus and live virus to demonstrate viral tropism for choroid plexus epithelial cells but little to no infection of neurons or glia. We find that infected cells are apolipoprotein- and ACE2-expressing cells of the choroid plexus epithelial barrier. Finally, we show that infection with SARS-CoV-2 damages the choroid plexus epithelium, leading to leakage across this important barrier that normally prevents entry of pathogens, immune cells, and cytokines into cerebrospinal fluid and the brain.