We model Bitcoin as a platform that intermediates between users and computer servers (“miners”) which operate the Bitcoin payment system (BPS), and studies the novel market design of this owner-less platform. We find that the BPS can eliminate inefficiencies due to market power, but incurs other costs. Having fixed transaction processing capacity, the BPS experiences service delays which motivate users to pay for service priority. Free entry implies that miners cannot profitably affect the level of fees paid by users. The paper derives closed form formulas of the fees and waiting times and studies their properties; compares pricing under the BPS to that under a traditional payment system operated by a profit maximizing firm; and suggests protocol design modification to enhance the platform’s efficiency.