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Analysis of absence of arbitrage normally ignores payoffs in states to which the agent assigns zero probability. We extend the fundamental theorem of asset pricing to the case of 'no empty promises' in which the agent cannot promise arbitrarily large payments in some states. There is a superpositive pricing rule that can assign positive price to claims in zero probability states important to the market as well as assigning positive prices to claims in the states of positive probability. With continuous information arrival, no empty promises can be enforced by shutting down the agent's subsequent investments once wealth hits zero. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.