The coronavirus disease (COVID-19) caused by SARS-CoV-2 is creating tremendous human suffering. To date, no effective drug is available to directly treat the disease. In a search for a drug against COVID-19, we have performed a high-throughput X-ray crystallographic screen of two repurposing drug libraries against the SARS-CoV-2 main protease ($M^{pro}$), which is essential for viral replication. In contrast to commonly applied X-ray fragment screening experiments with molecules of low complexity, our screen tested already approved drugs and drugs in clinical trials. From the three-dimensional protein structures, we identified 37 compounds that bind to $M^{pro}$. In subsequent cell-based viral reduction assays, one peptidomimetic and six non-peptidic compounds showed antiviral activity at non-toxic concentrations. We identified two allosteric binding sites representing attractive targets for drug development against SARS-CoV-2.