Nanopore sequencing promises long read-lengths and single-molecule resolution, but the stochastic motion of the DNA molecule inside the pore is, as of this writing, a barrier to high accuracy reads. We develop a method of statistical inference that explicitly accounts for this error, and demonstrate that high accuracy (>99%) sequence inference is feasible even under highly diffusive motion by using a hidden Markov model to jointly analyze multiple stochastic reads. Using this model, we place bounds on achievable inference accuracy under a range of experimental parameters.