We have used the Robert C. Byrd Green Bank Telescope to perform the most sensitive search to date for neutral atomic hydrogen (HI) in the circumstellar envelope (CSE) of the carbon star IRC+10216. Our observations have uncovered a low surface brightness HI shell of diameter ~1300 (~0.8 pc), centered on IRC+10216. The HI shell has an angular extent comparable to the far ultraviolet-emitting astrosphere of IRC+10216 previously detected with the GALEX satellite, and its kinematics are consistent with circumstellar matter that has been decelerated by the local interstellar medium. The shell appears to completely surround the star, but the highest HI column densities are measured along the leading edge of the shell, near the location of a previously identified bow shock. We estimate a total mass of atomic hydrogen associated with IRC+10216 CSE of M_HI~3x10e-3 M_sun. This is only a small fraction of the expected total mass of the CSE (<1%) and is consistent with the bulk of the stellar wind originating in molecular rather than atomic form, as expected for a cool star with an effective temperature T_eff<~2200 K. HI mapping of a 2 deg x 2 deg region surrounding IRC+10216 has also allowed us to characterize the line-of-sight interstellar emission in the region and has uncovered a link between diffuse FUV emission southwest of IRC+10216 and the Local Leo Cold Cloud.