  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.
