Nursing homes and other long term-care facilities account for a disproportionate share of COVID-19 cases and fatalities worldwide. Outbreaks in U.S. nursing homes have persisted despite nationwide visitor restrictions beginning in mid-March. An early report issued by the Centers for Disease Control and Prevention identified staff members working in multiple nursing homes as a likely source of spread from the Life Care Center in Kirkland, Washington to other skilled nursing facilities. The full extent of staff connections between nursing homes---and the crucial role these connections serve in spreading a highly contagious respiratory infection---is currently unknown given the lack of centralized data on cross-facility nursing home employment. In this paper, we perform the first large-scale analysis of nursing home connections via shared staff using device-level geolocation data from 30 million smartphones, and find that 7 percent of smartphones appearing in a nursing home also appeared in at least one other facility---even after visitor restrictions were imposed. We construct network measures of nursing home connectedness and estimate that nursing homes have, on average, connections with 15 other facilities. Controlling for demographic and other factors, a home's staff-network connections and its centrality within the greater network strongly predict COVID-19 cases. Traditional federal regulatory metrics of nursing home quality are unimportant in predicting outbreaks, consistent with recent research. Results suggest that eliminating staff linkages between nursing homes could reduce COVID-19 infections in nursing homes by 44 percent.