What is the effect of Central Bank Digital Currency (CBDC) on financial stability? We answer this question by studying a model of financial intermediation with an endogenously determined probability of a bank run and a remunerated CBDC that provides consumers with an alternative to bank deposits. Consistent with concerns among policymakers, higher CBDC remuneration raises bank fragility by increasing consumers’ withdrawal incentives. However, it also induces the bank to offer more attractive deposit contracts in an effort to retain funding, which reduces fragility. Accordingly, the overall relationship between bank fragility and CBDC remuneration is U-shaped. We evaluate the effects of different policy proposals aimed at reducing the financial stability implications of CBDC, and study extensions that allow for imperfect competition in deposit markets and bank risk-taking.