We develop a model of financial intermediation with remunerated Central Bank Digital Currency (CBDC) as consumers’ alternative to bank deposits and an endogenous risk of bank runs. Echoing widespread concerns, higher CBDC remuneration raises bank fragility by increasing consumers’ withdrawal incentives. On the other hand, it also induces banks to offer more attractive deposit contracts in order to retain funding, thereby reducing fragility. This results in a U-shaped relationship between bank fragility and CBDC remuneration. We evaluate policy proposals aimed at mitigating the financial-stability risks of CBDC, such as holding limits and contingent CBDC remuneration.