A central bank digital currency, or CBDC, may provide an attractive alternative to traditional demand deposits held in private banks. When offering CBDC accounts, the central bank needs to confront classic issues of banking: conducting maturity trans- formation while providing liquidity to private customers who suffer “spending” shocks. We analyze these issues in a nominal version of a Diamond and Dybvig (1983) model, with an additional and exogenous price stability objective for the central bank. While the central bank can always deliver on its nominal obligations, runs can nonetheless occur, manifesting themselves either as excessive real asset liquidation or as a failure to maintain price stability. We demonstrate an impossibility result that we call the CBDC trilemma: of the three goals of efficiency, financial stability (i.e., absence of runs), and price stability, the central bank can achieve at most two.