We demonstrate experimentally the feasibility of a two-state quantum bit commitment protocol, which is both concealing and partially binding, assuming technological limitations. The security of this protocol is based on the lack of long-term stable quantum memories. We use a polarization-encoding scheme and optical fiber as a quantum channel. The measurement probability for the commitment is obtained and the optimal cheating strategy demonstrated. The average success rates for an honest player in the case where the measurements are performed using equal bases are 93.4%, when the rectilinear basis is measured, and 96.7%, when the diagonal basis is measured. The rates for the case when the measurements are performed in different bases are 52.9%, when the rectilinear basis is measured, and 55.4% when the diagonal basis is measured. The average success rates for the optimal cheating strategy are 80% and 73.8%, which are way below the success rates of an honest player. Using a strict numerical validity criterion, we show that, for these experimental values, the protocol is secure.