Fast magnetic reconnection occurs in nearly all natural and laboratory plasmas and rapidly releases stored magnetic energy. Although commonly studied in fully ionized plasmas, if and when fast reconnection can occur in partially ionized plasmas, such as the interstellar medium or solar chromosphere, is not well understood. This Letter presents the first fully kinetic particle-in-cell simulations of partially ionized reconnection and demonstrates that fast reconnection can occur in partially ionized systems. In the simulations, the transition to fast reconnection occurs when the current sheet width thins below the ion-inertial length in contrast to previous analytic predictions. The peak reconnection rate is ≥0.08 when normalized to the bulk Alfvén speed (including both ion and neutral mass), consistent with previous experimental results. However, when the bulk Alfvén speed falls below the neutral sound speed, the rate becomes system size dependent. The normalized inflow velocity is ionization fraction dependent, which is shown to be a result of neutral momentum transport. A model for the inflow is developed which agrees well with the simulation results.