We analyze the run risk of fiat-backed stablecoins by uncovering a fundamental dilemma between stablecoins’ price stability and financial stability. We show that panic runs exist even though general investors only trade stablecoins in secondary markets with flexible prices. This is because stablecoins engage in liquidity transformation and the fixed price at which arbitrageurs redeem stablecoins for cash from the issuer in the primary market reinstates run incentives. We find that, surprisingly, stablecoins only authorize a concentrated set of arbitragers, which seems at odds with their proposed goals of price stability and decentralization. We show that this fact can be rationalized by centralized arbitrage acting like a firewall between primary and secondary markets. This centralization of arbitrage mitigates run-risk but exacerbates secondary market price dislocations, implying a tradeoff between stablecoins’ price stability and financial stability. We collect a novel dataset on stablecoin redemptions, trading, and reserve assets to calibrate our model. For the largest stablecoin, Tether (USDT), our estimates imply a run probability of 17.04% in September 2021 which decreases to 3.45% in March 2022. Such run risk would lead to broader financial stability risks, such as the fire sales of US dollar assets like bank deposits, Treasuries, and corporate bonds.