B cells are critical for the production of antibodies and protective immunity to viruses. Here we show that patients infected with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) who develop coronavirus disease 2019 (COVID-19) display early recruitment of B cells expressing a limited subset of IGHV genes, progressing to a highly polyclonal response of B cells with broader IGHV gene usage and extensive class switching to IgG and IgA subclasses with limited somatic hypermutation in the initial weeks of infection. We identify convergence of antibody sequences across SARS-CoV-2-infected patients, highlighting stereotyped naive responses to this virus. Notably, sequence-based detection in COVID-19 patients of convergent B cell clonotypes previously reported in SARS-CoV infection predicts the presence of SARS-CoV/SARS-CoV-2 cross-reactive antibody titers specific for the receptor-binding domain. These findings offer molecular insights into shared features of human B cell responses to SARS-CoV-2 and SARS-CoV.