Fundamental primitives such as bit commitment and oblivious transfer serve as building blocks for many other two-party protocols. Hence, the secure implementation of such primitives are important in modern cryptography. In this work, we present a bit commitment protocol which is secure as long as the attacker's quantum memory device is imperfect. The latter assumption is known as the noisy-storage model. We experimentally executed this protocol by performing measurements on polarization-entangled photon pairs. Our work includes a full security analysis, accounting for all experimental error rates and finite size effects. This demonstrates the feasibility of two-party protocols in this model using real-world quantum devices. Finally, we provide a general analysis of our bit commitment protocol for a range of experimental parameters. ; Comment: 21 pages (7 main text +14 appendix), 6+3 figures. New version changed author's name from Huei Ying Nelly Ng to Nelly Huei Ying Ng, for consistency with other publications