The mean free path of ionizing photons, 𝜆 mfp , is a key factor in the photoionization of the intergalactic medium (IGM). At 𝑧 5, however, 𝜆 mfp may be short enough that measurements towards QSOs are biased by the QSO proximity effect. We present new direct measurements of 𝜆 mfp that address this bias and extend up to 𝑧 ∼ 6 for the first time. Our measurements at 𝑧 ∼ 5 are based on data from the Giant Gemini GMOS survey and new Keck LRIS observations of low-luminosity QSOs. At 𝑧 ∼ 6 we use QSO spectra from Keck ESI and VLT X-Shooter. We measure 𝜆 mfp = 9.09 +1.28 −1.60 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 𝜆 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.