The mean free path of ionizing photons, $lambda_{rm mfp}$, is a key factor in the photoionization of the intergalactic medium (IGM). At $z gtrsim 5$, however, $lambda_{rm mfp}$ may be short enough that measurements towards QSOs are biased by the QSO proximity effect. We present new direct measurements of $lambda_{rm mfp}$ that address this bias and extend up to $z sim 6$ for the first time. Our measurements at $z sim 5$ are based on data from the Giant Gemini GMOS survey and new Keck LRIS observations of low-luminosity QSOs. At $z sim 6$ we use QSO spectra from Keck ESI and VLT X-Shooter. We measure $lambda_{rm mfp} = 9.09^{+1.62}_{-1.28}$ proper Mpc and $0.75^{+0.65}_{-0.45}$ proper Mpc (68% confidence) at $z = 5.1$ and 6.0, respectively. The results at $z = 5.1$ are consistent with existing measurements, suggesting that bias from the proximity effect is minor at this redshift. At $z = 6.0$, however, we find that neglecting the proximity effect biases the result high by a factor of two or more. Our measurement at $z = 6.0$ falls well below extrapolations from lower redshifts, indicating rapid evolution in $lambda_{rm mfp}$ over $5 < z < 6$. This evolution disfavors models in which reionization ended early enough that the IGM had time to fully relax hydrodynamically by $z = 6$, but is qualitatively consistent with models wherein reionization completed at $z = 6$ or even significantly later. Our mean free path results are most consistent with late reionization models wherein the IGM is still 20% neutral at $z=6$, although our measurement at $z = 6.0$ is even lower than these models prefer.
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